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00:00:04
good morning everyone00:00:07
welcome you now let's check that00:00:10
it all started with us00:00:14
dust2 is great00:00:17
let's see what we have there00:00:23
So00:00:26
clearly Vitalievich and Maxim Olegovich what00:00:30
it started so it started like that so yeah00:00:34
Maxim Olegovich is also ready to submit everything00:00:36
great so hello everyone00:00:40
good morning ladies of course I can00:00:43
say good morning and it will be true00:00:47
hello hello hello00:00:48
ok let me00:00:51
I'll tell you what's going to happen00:00:55
first of all first of all guys right away00:00:59
I reassure everyone there will be no recording00:01:01
We're worried, thank you all for coming00:01:04
first of all, who will watch the recordings too00:01:07
I'm glad to welcome you00:01:10
The duration of our webinar is almost00:01:13
12 hours as Maxim Olegovich said I00:01:16
listened to him well on his webinar there00:01:18
we have to finish around 11:55 here00:01:21
we will do so accordingly00:01:24
like this00:01:27
means a00:01:33
all tasks like Maxim Olegovich and00:01:34
we're off to a great start, let's do it as usual00:01:36
yes guys who solves the problem I think that00:01:38
Maxim Olegovich or even I’m ready from00:01:40
this task touches his pocket00:01:42
here is Maxim Olegovich in the chat00:01:46
let's decide like this00:01:51
first of all there is a schedule and ours00:01:55
broadcast it is divided into 4 blocks according to00:01:59
three hours in the description under the stream all00:02:02
there is information yes so of course00:02:06
Be sure to watch all twelve00:02:08
no hours needed someone ready someone00:02:12
really understands the beauty there00:02:16
number theories understand that they will need00:02:18
the latest topics for someone to prepare for00:02:22
this is not technical olympiads at all00:02:25
I really need it and I will be there urgently00:02:27
recommend 4 hours when it starts00:02:30
the last one there are two blocks go to00:02:33
webinar by Anna Vladimirovna00:02:34
By the way, who hasn't seen it yet Vladimir on00:02:36
spends today in a child according to parameters00:02:39
so friends00:02:41
I’ll send you the link, plus on the channel yourself00:02:44
you can find it here00:02:46
new started unclear wording00:02:49
yes yes yes yes yes yes yes well what is what00:02:51
that's all let's do it hard00:02:57
Please don't spam in chat00:02:59
it’s clear that for now yes, but as soon as we00:03:02
let's start chatting let's ask a question00:03:05
essentially ask here00:03:09
for the highest standard yes yes for the highest standard00:03:13
I'm afraid that I don't just have to sit until00:03:16
the end is still needed after the webinar00:03:19
solve problems that will remain the same00:03:21
there will definitely be Maxim Olegovich00:03:23
Seems00:03:25
I posted a piece of paper, now I'll check it out00:03:28
description he is00:03:30
In general, I’ll look at the description to understand yes00:03:33
yes yes yes yes here is the first link00:03:35
no one has material for the webinar00:03:36
displayed just refresh the page00:03:39
do this and I'll be very happy00:03:41
I would appreciate it if you thank me00:03:44
like me comments00:03:46
in general there seems to be so much free stuff00:03:49
content for you I think that00:03:53
we will deserve a generous like from you so well00:03:59
that don't forget don't forget so00:04:03
now I'll quickly00:04:06
I'll see if there are any questions in the chat00:04:09
it is essential that you can immediately answer and00:04:11
we will start our math00:04:15
open the window and I will listen to both of you00:04:19
brilliant everything is in the description yes yes yes00:04:22
Yes Yes Yes00:04:23
so who by the way will watch to the end00:04:27
let's plus to chat plus to chat record00:04:31
I haven't already talked about this00:04:33
worry about who will finish00:04:37
let's make not new current minuses00:04:40
pluses only pluses00:04:43
that's all you made a promise I'll check00:04:46
I'll check if there's a plus sign at some point00:04:49
will become smaller00:04:50
I'll calculate the pipe and figure it out00:04:53
So00:04:56
okay let's start the theory has begun00:05:01
numbers today I chose topics that00:05:07
as I usually say at theory webinars00:05:09
Numbers Olympiad has two big themes00:05:11
these are the remainders and factorization we00:05:15
today we are talking about leftovers comparison by00:05:18
modulo quadratic residues antiderivatives00:05:21
Chinese roots about leftovers that's all00:05:23
miracle about factorization we00:05:26
We won't talk too much today00:05:30
let's leave this for some maybe00:05:33
the next webinars here and how those00:05:38
who is really first for bot games00:05:41
round is more than a real thing00:05:43
Problem 19 plus probably part00:05:47
olympiadists will know everything00:05:50
Now there's a lot going on, I thought a lot00:05:53
some of the Olympiads but in principle00:05:55
repeating it isn’t even harmful, especially for now00:05:57
let's rock some simple ones00:05:59
repetition tasks are all good00:06:02
I'll eat when I'm ready too00:06:04
watch during the break so you understand00:06:06
A00:06:07
Here we have a round of 3 hours in the middle00:06:11
short 5 minute00:06:14
not to answer any of your questions00:06:17
according to the stream, you should go pour me some tea there00:06:21
there's still a little more to do between rounds00:06:24
more 5-10 minutes plus in between00:06:28
some second and third round there00:06:31
There will already be a break, like 10-15 minutes00:06:35
Here00:06:36
that's it, let's kind of pay tribute00:06:41
there was no respect for sharks in sushi00:06:44
here we go and let's begin00:07:05
food a00:07:09
first blog dedicated to00:07:12
modulo doubt00:07:14
small zafer mothers wilson plus00:07:17
Let's talk more about their different uses00:07:20
that is, we will solve problems on this and not00:07:22
one must think that number theory means00:07:24
I will only tell you the theory of the problem00:07:26
we are also kind of going to decide in general00:07:28
speaking, let's start with a very simple thing00:07:33
such as division with remainder00:07:36
you and I all understand perfectly well that00:07:37
numbers can be divided with stark by each other00:07:39
friend, but for example, I’ll divide the number00:07:42
26 with a remainder of 4 what is it is 400:07:49
multiply by 6 plus 200:07:53
well there is division with a remainder00:07:55
certain rules the remainder must be00:07:59
not more00:08:01
than the divisor00:08:03
less very strictly less than the divisor00:08:06
And00:08:09
not less than 000:08:11
Well, as we well know, we share with00:08:14
the rest doesn't have many nice properties00:08:16
well, for example00:08:18
the balances do not add up but with each00:08:23
number 26 remainder of the 2nd number 27 and remainder00:08:28
3 their sum has a remainder that is not 2 plus 300:08:32
because it will be more than00:08:34
four would already be one00:08:36
it is clear that they are also bad00:08:39
multiplied every time you need like00:08:41
recalculate the balance again00:08:44
divide by the divisor like this it's not00:08:47
very convenient thing00:08:49
from those yes yes at the very end at the very end00:08:52
that's where the joke starts00:08:56
like that00:08:59
so I would like to come up with some00:09:02
a more convenient thing than dividing with00:09:05
the rest of it came up with this comparison by00:09:08
module, that's what Maxim likes to say00:09:11
Olegovich we put on glasses modulo what is it00:09:16
such a designation a is comparable to b by00:09:20
module m00:09:21
they write differently sometimes they write in brackets00:09:24
fashion m00:09:29
sometimes they write a module at the top, it’s like00:09:33
do you like it better this way and this way00:09:35
they will understand00:09:37
this designation is equivalent to that00:09:40
the difference is divided by m00:09:45
then no no not at the end of Physics and Technology will be00:09:49
calm down it's okay what you are00:09:50
you're worried00:09:52
will be everywhere00:09:54
now actually00:09:57
that is, if you are preparing for one of those00:10:00
really, the first three hours before this is it00:10:03
what would you be interested in, well, it’s there00:10:06
needed at such a basic level at physics and technology00:10:08
number theory is not particularly difficult00:10:12
So00:10:16
for what class anyway?00:10:19
should be clear00:10:21
starting from 6 well for now he has a guy00:10:25
definition with remainder 1 block is complete00:10:28
can be understood by sixth to seventh graders00:10:31
here, but at the same time, if you are, as it were, before this00:10:34
not a bot the Olympics in the 10th 11th, well00:10:36
as we said in books 16 to 1900:10:39
tasks here is the first block santa00:10:43
So00:10:48
Daniel I don’t recognize you why don’t I00:10:51
I'll find out00:10:52
you because nickname only Daniil therefore00:10:56
I don't recognize you00:10:58
good means firstly such a property00:11:01
secondly, it’s probably not difficult to understand that00:11:03
this comparison tells us that00:11:05
numbers a and b give the same remainders00:11:10
give the same balances00:11:15
acute with00:11:18
business on m00:11:22
in general actually considering how much00:11:24
need to write letters00:11:26
to say what the numbers give00:11:29
the same remains are already here00:11:31
enough reason to come up with00:11:33
modulo comparison let me give you00:11:35
a few examples00:11:36
for example, 2 is comparable to five modulo00:11:39
300:11:42
7 comparable to00:11:45
12 modulo five00:11:50
minus 4 is comparable to two modulo 300:11:56
until I don’t recognize of course of course externally00:12:04
What are you guys looking at first?00:12:08
don't worry about what you don't have00:12:12
the remainder from none of the parts, that is, here00:12:14
I am writing about comparing 7 cm with 12 modulo00:12:17
five and no problems whatsoever00:12:20
one of the numbers is not a remainder00:12:21
I don’t have a definition for an A, yes00:12:24
formally how to check what it is00:12:26
correct comparison take write00:12:28
difference check what is divisible in a given00:12:31
in this case a three in this case a five and00:12:33
here is the most interesting case with00:12:35
negative numbers for them all too00:12:37
it works just in case, I remind you that00:12:40
firstly when you calculate the balance00:12:41
negative numbers it should be00:12:44
non-negative that is, if I drink00:12:46
counts the remainder of the number -4 definitions00:12:49
sodium must be multiplied by minus here00:12:50
deuce but also add two further00:12:55
it happens that the rest are 2 guys00:13:01
we all started to get sick so let's all00:13:03
let's delve into this more00:13:05
understand because I would like00:13:08
understand at what level00:13:11
to tell that is, it is clear that the first00:13:14
block I tell you in more detail it for00:13:16
there will be more to a wider audience00:13:20
not so detailed, that is, those who want00:13:22
what is not hard first you have00:13:24
a sheet of paper with tasks then you can00:13:27
open it and calmly decide from there00:13:29
tasks even the same tasks on the topic00:13:31
Euler is already being delivered by some00:13:35
difficulties00:13:36
why is -1 not a remainder by definition?00:13:40
by definition, the remainder is less than the divisor00:13:43
and not less than 000:13:47
hello paul00:13:51
aiming at 10:02 somewhere00:13:55
this is what she will really give00:13:58
deuce but00:13:59
in such cases there is no need to count00:14:01
the rest to check what the comparison is00:14:04
done you can just take00:14:06
subtract the first number from the second or 2 s00:14:09
1 is not important and check whether it is divisible or not00:14:12
shares00:14:13
here it is00:14:15
with the negative one more time, here I am00:14:17
I just repeated it one more time but00:14:20
negative numbers why they us00:14:22
should be confusing at all, I don’t see a single one00:14:25
reasons we can consider the differences00:14:28
Can00:14:30
minus 4 minus 2 is minus 6 minus00:14:34
six is divisible by three so it is00:14:36
the comparison is also true00:14:39
figured it out00:14:43
Maybe00:14:45
as belonging to natural or whole with00:14:46
plus, well, whole with a plus, that is, the remainder00:14:50
0 there may well be one more time when00:14:53
I'm talking about the leftovers, but here we are00:14:55
if there were negative balances we should00:14:57
understand that they don’t exist00:14:59
but at the same time for a negative number I00:15:01
I can count the rest, I have nothing00:15:03
will interfere if I want to be what I am00:15:05
negative number like there00:15:10
-11 divide the remainder by 4 like this00:15:13
works I need to find the nearest number00:15:17
which is less than or equal to -1100:15:20
which is divisible by four is minus 1200:15:24
minus 12 divided by four we get 400:15:28
multiply by minus 3 and add the remainder00:15:32
the remainder will be one00:15:36
sorted it out yes00:15:38
For some reason there is a lot of confusion here00:15:41
what type -11 must have00:15:43
the same remainder as 11 in no case00:15:46
in no case did we figure it out in an hour00:15:48
it works like this00:15:50
that's when we talk about comparisons00:15:54
everything is even easier for the module00:15:56
so because they just calculated the difference00:15:58
let's guys move on like this00:16:01
but we would like to cover more00:16:05
there are problems here00:16:08
you can00:16:09
just practice with these yourself00:16:13
comparisons to make sure that yes, well, as if with00:16:16
negative numbers no problem00:16:18
does not occur00:16:19
so well, it’s good to compare modulo if00:16:23
if they were only needed for one00:16:24
functions00:16:26
simplify the notation for us, probably this00:16:29
it would be nice but not so great00:16:31
so let's talk about them00:16:33
We'll see00:16:36
on the properties of comparisons00:16:42
what properties does comparison have on00:16:45
just in case the black guys follow the link00:16:47
All tasks are in the description, that is00:16:50
It’s better to perceive them not just by ear00:16:53
look at the first page we are still with00:16:56
it's like they started00:16:59
You can take negative rates00:17:02
add to the module and get00:17:03
correct balances yes it's true but how00:17:08
it would be better once more right away and00:17:11
consider that is you00:17:13
find the number that is divisible by00:17:17
divisor not exceeding the given one for00:17:20
the numbers minus 11 are exactly -12 not -800:17:24
when we divide by four00:17:26
and so properties of comparisons properties00:17:32
comparisons00:17:34
firstly, comparisons can be added here00:17:38
and it’s not very good to stack bets, but00:17:40
because we need to recalculate them for00:17:43
everyone, well, check whether they succeeded00:17:46
you so the amount00:17:48
the balances became greater than or equal to00:17:50
divisor but for comparison everything is ok00:17:53
how can this be written if a is compared with00:17:57
b modulo mc is comparable to d modulo m00:18:02
therefore a + b00:18:04
comparable to c + d modulo m00:18:14
yes, well, in terms of properties, this is the very first00:18:17
the fact that they purely give the same rates but00:18:19
I think this is already clear yes my00:18:21
the principles have already said this because well00:18:24
ok if we subtract one from the other00:18:26
the remains were different then the remains were not00:18:28
would decrease and accordingly it would00:18:31
shared still it's very much not shared00:18:34
that's why it's perth but the properties behind it00:18:38
zero let's start with the first why leave00:18:41
can you be more precise why the comparison is possible00:18:43
fold00:18:47
that’s not what to do, I wrote yes thank you00:18:50
designation00:18:52
why comparison can be added00:18:56
let's pump the division formally purely00:18:59
I will prove by definition that I00:19:02
must prove and must prove that00:19:04
difference a + c minus b plus d00:19:07
divided by m00:19:13
so let's regroup a little00:19:16
differently00:19:17
let's write a minus b00:19:20
+ c minus d divided by m and00:19:29
so why is this not true yet?00:19:33
we know how we will understand why it is obvious00:19:36
minus b is divisible by m because a00:19:38
comparable to b in modulus but simply in00:19:41
definition it turns out that the difference00:19:42
are divisible by m00:19:44
Well, minus d is also divisible by00:19:48
everything is great we got two numbers each00:19:51
of which is divided by m their sum is divided00:19:53
on m therefore we have proved divisibility00:19:58
obviously yes00:20:00
the second property is that you can subtract00:20:05
yeah, I don’t know if we want it separately00:20:07
highlight that is, you can write00:20:10
plan let's highlight it means if a00:20:13
comparable with b modulo m.c. comparable to d00:20:17
modulo m it turns out00:20:21
difference a minus c00:20:24
comparable to b minus d00:20:27
modulo m00:20:30
the evidence is exactly the same00:20:33
subtract and regroup very much00:20:36
I recommend doing it as00:20:37
exercises if you haven’t seen them before00:20:40
compared to module00:20:45
these anvar yes we are here in space or00:20:48
brackets regrouped00:20:50
please note this is the same thing00:20:52
the expression is simply in this form as if not00:20:55
it is quite obvious why why they are divided into00:20:56
m in this form is obvious00:21:00
that every two terms are divisible by00:21:03
I00:21:05
Can add and subtract well00:21:09
standing we usually know how to do if we have00:21:12
it says equality, that's what convenience is00:21:14
modulo comparisons for some time now we00:21:17
we can simply perceive this as00:21:19
equality is the most important thing guys right away00:21:22
draw your attention00:21:23
the module should be the same00:21:26
the number m must be the same00:21:30
in the second block we will talk about what00:21:32
occurs in different modules and00:21:35
we will prove there to the Chinese prisons about the remains00:21:38
let's solve problems for it00:21:40
yes I really want to multiply yes00:21:43
that is, I want for example00:21:47
be able to do the following, come on00:21:50
berths simple let's start from the beginning00:21:52
let's learn how to multiply by a constant if a00:21:56
comparable to b modulo m00:21:59
as a whole00:22:02
then I want to say that a k is comparable to bk00:22:06
modulo m00:22:10
why is this true? let's do it again00:22:13
we'll write the difference, look, I want to prove it00:22:15
that arch minus b k is divided by m00:22:19
I take out k I get k by a minus b is divided00:22:22
us00:22:25
Well of course it is divided to where it is00:22:28
gets away from us a minus b is divided by m00:22:32
because from the definition comparison a00:22:35
minus b divided by m we multiply by00:22:38
integer means divisible00:22:41
can be flipped to the right or to the left00:22:43
Of course, who will forbid us and who will you?00:22:46
will prohibit transferring to another part00:22:50
so with this property we got to everything00:22:54
even easier00:22:56
4 properties why you can multiply00:22:58
doubt that is, not simply multiplied by00:23:01
a constant, namely, multiply a in it00:23:05
that is, why from the fact that the bridge is guarded00:23:08
bc is comparable to d we can write that fathers00:23:11
compare sbd modulo m00:23:16
this is already four properties this is this00:23:19
already more cunning properties now we have it00:23:21
we'll prove it here00:23:29
How00:23:30
choice now let's for what are you doing00:23:32
there in the comments00:23:38
Igor, as it were, the next thing to decide, yes, that is00:23:43
as if in this half hour I will repeat that00:23:47
what was on the course compared to00:23:49
module because the theory is absolutely there00:23:51
the same and as if I'm not now00:23:54
I’ll say further it won’t be clear at all00:23:57
so that's why wait patience00:24:01
while turning, wait, be patient00:24:03
what am I talking about please be patient00:24:06
you can solve problems00:24:08
from the 2nd leaflet he is no longer familiar to you00:24:14
so there was a question there00:24:19
that there was a question about the world writing something with the anus00:24:23
to the world I answered yes hello00:24:25
Gregory00:24:31
like this when dividing I haven’t got anything yet00:24:34
told you let's not rush, don't rush00:24:38
we're not in a hurry, it's not just for minus00:24:42
works for and for the pros00:24:44
where where let's jump please00:24:49
if you ask a question, then a little more00:24:51
specifically00:24:56
so some I hope that how would you00:25:00
either joke or just come to us00:25:03
joined if so then00:25:04
reconsider the moment when I started00:25:07
modulo straightening seems to be good00:25:10
let's not repeat it now00:25:12
literally rewind 10 minutes and00:25:16
so we will prove this property00:25:19
it is more difficult to prove, look00:25:21
I need to get my father to compare their b.d.00:25:25
if I write00:25:27
the difference is that I will succeed and that is minus00:25:31
b d if I wrote a minus b and here00:25:34
c minus d and somehow multiplied them for me00:25:36
there would be a lot of extra stuff out there, yes I have00:25:39
it would turn out to be 4 terms bad therefore00:25:43
we do so we will use00:25:45
the third point is that if a00:25:48
comparable to b modulo m00:25:52
Well, it follows that the father is comparable from bc00:25:57
modulo m00:26:05
so look at the mazda level, here is the army already00:26:10
got it comparable to bc let me00:26:13
I will get bc in this comparison if c00:26:17
comparable to d modulo m00:26:20
then it follows that cb00:26:24
comparable to00:26:27
d.b. or better yet, I’ll write the prices and save them00:26:30
sbd00:26:32
what else did I do I just multiplied00:26:35
on b this comparison can be done by00:26:37
third property about00:26:42
look how it turned out father compare with00:26:45
bc and bc are comparable to b.d. well of course it is00:26:49
means that the estimate is comparable sbd00:26:52
because father and bc are the same00:26:55
remains of killed b.d. means from the father and b d00:26:58
are also the same residues, which means they are00:27:00
comparable in modulus and00:27:06
so there will be questions and answers00:27:11
and now you can ask how you see me00:27:13
I answer them00:27:15
the first ones will be well in 1 through to00:27:17
relax and questions and not related there00:27:20
with what I told and are now arising00:27:23
I’d rather ask them questions now00:27:26
hello Stanislav00:27:28
So00:27:31
over here let's still ask you all00:27:34
Is it clear from the fourth property00:27:38
let's put a plus sign there00:27:41
prophylactic00:27:42
once again please about multiplication00:27:45
let's let's then okay but look00:27:50
from the fact that by opening fb it follows that00:27:53
from the test in it sbt simply multiply by c00:27:56
from the fact that from the country Mazda modulo m00:27:59
it follows that bts touch sbd I just00:28:01
multiplied by b00:28:03
see here from cbc here bcd if00:28:07
let's remove sbc, it means they have the same00:28:09
their balances are the same balances bc and00:28:12
here and there the same means these numbers00:28:14
also the same remains, so we sorted it out00:28:17
That00:28:19
Great00:28:21
now let's then00:28:23
let's talk about00:28:25
exponentiation00:28:27
the last property that made me00:28:29
interested00:28:30
I didn’t write others on the leaves00:28:33
this is exponentiation let's do this00:28:36
we'll do it here00:28:39
So00:28:41
5 properties00:28:43
if a is compared with b00:28:46
modulo00:28:48
m.t. natural00:28:52
then a to the power k is comparable to b to the power00:28:56
k modulo m00:29:00
well, yes, and you call it induction, as it were00:29:05
can be formally proven by00:29:07
induction actually but watch you00:29:09
there is a comparison of osago nim fb modulo00:29:11
m&a is comparable to b modulo m, let's do it00:29:15
we will multiply them multiply we already00:29:17
we can do this is the fourth property that we00:29:20
just proved00:29:21
then a square is comparable to b square by00:29:24
module m00:29:29
when dividing yes yes yes00:29:37
ok well let's get it in squares00:29:41
cubes we get us there is this comparison00:29:43
we have this comparison, let's give them00:29:45
let's multiply00:29:47
we get a cube comparable to 2 cube modulo00:29:50
m00:29:52
and so on, of course it’s possible00:29:55
formulate in a neat induction but on00:29:57
in fact, and without such a terrible word00:30:00
everything is clear, we multiply the left side00:30:03
multiplying the right side turns out to be a in00:30:06
4 compare size 4 everything is easy and simple00:30:11
Here00:30:14
with non-integer powers just forget00:30:18
just forget that there are not whole00:30:21
I'll explain why00:30:23
well imagine what you had00:30:25
comparison 4 is comparable to nine modulo00:30:28
five and00:30:29
you are so confident you want to extract00:30:32
the root, that is, raised to the power 1 200:30:36
take the root and get 2 comparable to00:30:39
three modulo five00:30:40
nonsense pooping is actually completely00:30:44
it’s clear why this happened, the thing is00:30:46
that nine is not only three in00:30:49
squared from also minus three in00:30:50
squared and here with minus three is two00:30:53
really comparable but only us00:30:57
where do we know what to write plus or minus00:30:59
describe plus or minus what one of these00:31:02
comparisons will be performed and why in00:31:05
what's the point then00:31:06
Here00:31:10
[music]00:31:11
so no, here's the maxim, let's go without these00:31:14
bytes about the advantages of the chat and who is waiting00:31:17
joke once again concentrated decides00:31:20
OK00:31:22
so great for green tea we answered00:31:27
in general, these are all the main properties00:31:30
which we will use for00:31:32
so that this is not something for us00:31:37
complex unusual let us a little bit00:31:39
Let's practice solving simple problems on00:31:42
this topic00:31:44
no, but Maxim one more time, you decide, you decide00:31:48
I already said yes, it’s like somewhere in00:31:52
11 we00:31:54
let's move on to the little farm theorem next00:31:57
there will be a mailer for you and we’ll decide00:31:59
tasks on terra mailer00:32:01
So00:32:04
everyone let's make room for you all00:32:08
properties are listed on leaflets therefore00:32:10
you will never lose them, but in general they00:32:13
should be here and here00:32:20
we still00:32:21
m how does it help solve real00:32:23
tasks00:32:34
between will be and in fact00:32:38
faster than you might think already00:32:41
on the first block00:32:42
will ask from the international but really00:32:45
quite ancient I won't do it00:32:48
very complex but formally funny00:32:52
all OK00:32:53
about division, come on, I promised00:32:57
say when dividing actually this00:32:58
it will be further useful to us when we are00:33:00
talk about little tower company already00:33:03
but let's just quickly understand that00:33:04
division doesn't work see 9 compare with00:33:08
six modulo 3 if I divide by00:33:10
three then it turns out three is comparable to00:33:13
two modulo 300:33:15
what00:33:18
not this way00:33:22
it's completely clear why this happens00:33:25
look 9 is comparable to six in modulus00:33:27
3 why00:33:28
the difference is 9 minus 6 I can rewrite it as00:33:31
3 times 3 minus 2 and00:33:34
it is divisible by 3 not because00:33:37
this difference is divided by three due to00:33:39
that this common factor is divisible by00:33:41
throne and00:33:42
when I shorten it naturally00:33:45
something terrible happens, gets lost00:33:48
divisibility which means comparison I write00:33:51
I can not00:33:52
in fact, sometimes you can divide00:33:55
divide if that by which you divide mutually00:33:57
just with the module but let's talk about it a little00:34:00
a little later, be patient while you just00:34:05
remember that we must share wisely00:34:09
so yeah00:34:12
to between00:34:14
like the Tibetan Olympics which somehow00:34:17
Maxim Olegovich I don’t remember what year00:34:19
already 1700 something00:34:22
Here00:34:24
two minutes to think before some00:34:27
I will naturally give tasks00:34:32
Danil right now there is no leaflet with00:34:35
Euler but in my opinion there is the last one or00:34:38
penultimate00:34:39
from this leaflet there is a mailer for you00:34:43
so good, let's practice00:34:46
first task before computer what is the number00:34:49
one thousand one thousand one00:34:52
per thousand 200:34:55
per thousand 300:34:58
minus 24 points a00:35:01
girl on00:35:03
999 yes, I don’t think I said that, but you00:35:07
did you understand that this icon is00:35:09
denotes divided by00:35:11
By the way, strangely enough, this is in Russia00:35:14
until accepted00:35:16
use offset divided by in00:35:19
foreign literature accepted00:35:21
use this icon a 9b and00:35:24
so sometimes it's quite hard to read00:35:28
foreign literature switch to00:35:30
this and something that will not separate into b a00:35:34
on the contrary, it will separate b, that is, b is divisible by00:35:37
but quite difficult to understand00:35:39
that's the thing, okay, we're not talking about that now00:35:42
will00:35:47
cabinets is divisible by a thousand 400:35:52
let's start with the point why they are divided by 99900:35:57
this is the work here we have to00:36:01
understand why there were comparisons at all00:36:03
modulo comparison allows us to00:36:08
expression, replace numbers with more convenient ones00:36:12
And00:36:13
other numbers that are comparable to00:36:16
the original ones, for example, are here00:36:19
number 100000:36:21
1000 divided by 999 is fear and horror00:36:25
why write thousands if you can write00:36:28
just one00:36:30
let's write it like this: 1000 is comparable to00:36:33
units and modulo 999 it is obvious00:36:37
Truth00:36:38
1001 is comparable to two modulo 999 thousand00:36:44
and 2 is comparable to three00:36:49
1300 I couldn’t 4 helped 999 so that’s it00:36:55
product of one thousand times one thousand one00:36:59
per thousand 200:37:01
per thousand 300:37:04
comparable to 1 to 2 to 3 to 4 helped for00:37:08
999 product of 12 by 3 by 4 is 2400:37:13
modulo 999 but accordingly when I00:37:17
and 24 subtracting 24 I get00:37:21
compared to zero modulo 999 means00:37:24
divisible by 99900:37:32
thief, look, here, in principle, you can00:37:36
it would be of it off the parks to do that00:37:39
it would seem the same thing, but what is wrong with us00:37:43
replaced a thousand with one, well, probably00:37:46
So it’s clear, yes, they replaced it with the remainder00:37:48
still ok yeah00:37:51
so what is the last arrival is unclear00:37:54
which one00:37:59
hopper00:38:01
speaks the absolute truth of course00:38:03
there cannot be negative residuals00:38:05
you can compare any numbers at least00:38:08
positive enough recipes with00:38:14
work00:38:17
Well, again guys, please ask for more details00:38:21
when you ask questions because I00:38:24
here the work is everywhere see more00:38:26
once the logic in what was before is not 100000:38:28
for 1000 one for 1000 2 for 3 I replace00:38:31
each number by its remainder when divided00:38:34
at 999 in the end the rest is all00:38:40
works the same00:38:43
formally, if you rewrite this00:38:46
completely descending to the properties of comparison00:38:51
modulo there are 4 of me00:38:54
comparisons 100:38:59
200:39:04
you are 2 more comparable to three00:39:08
100300:39:10
comparable to 4 modulo 999 I just them00:39:14
I multiply because I can00:39:17
but of course we need to perceive it differently00:39:20
we perceive it a little exactly as we do00:39:24
take the number and replace it with one that is convenient for us, otherwise00:39:29
that this can really be described00:39:31
through properties we just need it00:39:33
understand, we did this once00:39:35
then of course go down in detail00:39:38
We won’t get to the properties otherwise now00:39:40
let's finish our drink00:39:42
so we sorted everything out great and in the end00:39:45
accordingly, we found that it is comparable00:39:47
with four subtract 24 we get that00:39:50
the difference is comparable to zero, which means they divide00:39:52
to 99900:39:55
word and now yes indeed00:39:58
With00:40:02
point b when we talk about the module00:40:05
thousand modulo 1004 becomes everything00:40:08
somewhat more interesting00:40:10
the number 1000 itself generally gives00:40:13
remainder of 1000 definitions per 100400:40:17
well, like it didn’t get better 1000 is 100000:40:22
so this is where they really start00:40:26
work comparison by fashion on what00:40:28
a more convenient number can be replaced00:40:31
thousand when we talk about divisibility by00:40:33
24 to 1004 considering that the guys are in the chat00:40:38
we already wrote about this, let me here as00:40:40
I won't wait too long00:40:42
cat by the tail yes00:40:44
when we talk about divisibility by 100400:40:47
it's better to take minus four, but that's00:40:51
prime number less modulo less00:40:54
small-small is not scary and the number00:40:56
minus 4 is absolutely true00:41:02
1001 what can be replaced with00:41:05
-300:41:08
further by -2 and -100:41:12
instead of multiplying terrible00:41:15
scary numbers I can multiply minus00:41:17
4 to minus 3 to minus to -100:41:19
don’t forget to put the parenthesis of course00:41:26
the product is 2400:41:29
accordingly when I take away from him00:41:31
24 I get a number that is divisible by00:41:34
1004 record will be saved until00:41:37
minus, well, because with a plus it’s not true00:41:41
because 1000 minus minus 4 is 1004 and00:41:45
she shares you four more plus four00:41:49
this of course doesn't work00:41:52
before we explained for about 15 minutes00:41:56
back00:41:59
here you already understand what00:42:02
beauty modulo comparison as if00:42:05
we didn't know them, we would hardly have been able to00:42:08
replace negative numbers00:42:11
now we know how to do it we understand00:42:13
why is everything okay00:42:15
so this is a simple example00:42:18
Let's quickly look at a couple more00:42:21
so I won’t be in a row the whole sheet until00:42:26
I still have a goal to disassemble00:42:28
go at some point we lose a little00:42:31
firm hour we'll practice a little more00:42:33
and it will already be00:42:34
something more interesting00:42:37
so let's clear everything out here completely00:42:53
look where there is no commenting varta00:42:56
proposal to multiply by minus one a00:43:00
when to figure it out00:43:02
you show me what's wrong00:43:05
[music]00:43:06
replace thousand with -4 that's ok00:43:11
naturally when you know about00:43:13
modulo comparisons00:43:14
take and multiply for no reason00:43:17
each minus one is like nothing00:43:21
That's a cool idea, that's why of course00:43:24
locally some tasks this will help us00:43:26
solve but we are trying to build a theory00:43:29
which in general will be useful00:43:32
she's the target now00:43:36
in a sense there is a lasso00:43:39
moreover, there are even 5 literature00:43:43
about this but let's get to this a little00:43:47
a little later these metal ones00:43:50
I mean, there are some there, but not quite00:43:52
those that you may think00:43:55
geometry so00:43:59
well, the second task very quickly, let's00:44:02
let's do it00:44:03
that means all the points00:44:06
at once00:44:08
2018 4 2018 remainder when divided by00:44:12
three00:44:17
If00:44:19
but it has a remainder of 000:44:21
equivalent things of course of course point00:44:24
c now let's approach the function will be for us00:44:26
very useful when we start talking00:44:28
about the little farm theorem00:44:31
so far 4 2018 I want to calculate the balance00:44:35
What is the logic when dividing by three?00:44:38
let's replace the number 4 with a more convenient one00:44:41
the fact is that 4 is comparable to one00:44:45
module 3 in by the last property i00:44:49
I can raise to a power00:44:52
4 in 2018 is comparable to 1 in 2018 modulo 300:44:58
but this is one, which means the remainder is one00:45:04
point b00:45:06
There is00:45:07
in 2017 remainder when divided by 700:45:17
ours has also expired and can be replaced00:45:21
if we just counted the remainder then neither00:45:25
what from easy that is six where00:45:27
we'll get away from her but fortunately00:45:30
you can write 6 comparable to minus00:45:32
unit modulo 700:45:40
to minus one and now we raise00:45:44
degree 2017 it follows that 6 201700:45:49
this is the same as -1 in 2017 that is00:45:54
minus one modulo 700:46:00
so that is, I can write here -100:46:05
Please note we haven't solved the problem00:46:08
they ask us to find the rest, they don’t ask us00:46:11
find some number with which00:46:13
comparable 6 2017 we are asked to find exactly00:46:16
the rest, that is, now we have to be honest00:46:18
calculate the remainder but it is clear that the sediment00:46:23
600:46:24
you can just add seven remainder00:46:26
hasn't changed we got 6 and six years00:46:29
there really isn't any left anymore00:46:31
forget about such moments even if they are00:46:34
quite simple and00:46:36
this is the point of this task, yes he is already like this00:46:40
the fourth is undecided00:46:42
13th degree five hundred fifty five00:46:45
definitions for 900:46:48
the thing is that 13 how do I replace it with00:46:51
something can be replaced with a four or00:46:53
minus 5 minus 5 obviously won't be more00:46:56
comfortable yes for four nok for 400:46:59
we can replace this we understand and00:47:02
what's next00:47:07
if it was black everything would also work until00:47:10
there just wouldn't be a minus and that's all00:47:12
here are 4 to the power 555 agar great fedot00:47:17
programmer detective00:47:19
2 stars is clearly not something00:47:23
mathematical00:47:25
so look, let's just00:47:29
let's start counting stupidly00:47:31
degree 4 1 how is it comparable modulo 900:47:35
Well00:47:37
4 with what else now squared is 16 then00:47:42
there are seven people00:47:45
You can also replace it with minus two, but00:47:48
ok let's leave principle 7 for now00:47:51
not that important00:47:52
Let's count 4 cubed00:47:55
so this is 7 by 4 important point guys I00:48:00
I don't have to count 4 in a cube myself, I can count00:48:03
just multiply the previous comparison by00:48:06
four 7 by 400:48:08
2800:48:11
Well, that is, unit modulo 9 o00:48:15
look it turned out to be one00:48:19
let's predict what will happen next00:48:24
but we got one after one 4 400:48:28
video he will give the remainder 4 because we00:48:31
just ones and multiply by four00:48:36
these remnants will continue to go along00:48:40
cycle, that is, if we solved this00:48:44
problem without modulo comparison we would00:48:46
further explained why everything00:48:47
let's go in cycles anyway00:48:49
let's do it yes because I think it's not00:48:51
for everyone there is something really obvious why00:48:55
then there is a looping here we have00:48:57
the rest of 4 was repeated, the thing is that00:48:59
the remainder of the next number is unambiguous00:49:01
determined by the remainder of the previous one00:49:04
just put the previous balance on00:49:06
four and find a new remainder and there00:49:09
share with the machines that's why after00:49:13
a four will always be a seven00:49:15
after seven there will always be one after00:49:18
units 4k and so everything went in circles00:49:22
and then you will need to count there00:49:23
what is the number 555 in00:49:27
this cycle, well, basically00:49:30
This is a normal problem that can be solved00:49:34
what trick do they allow us to do00:49:36
modulo comparisons00:49:37
so we know that 4 cubed is comparable to00:49:42
unit modulo 9 a00:49:45
what remainder does 555 give when divided?00:49:49
for a three, let's make it 55500:49:52
three with remainder but remainder 0 5 plus 500:49:57
plus 5 15 based on divisibility by 300:49:59
the sum of the numbers is divisible by 3, so it will be00:50:02
the remainder is 0, well, you can even divide it fairly00:50:04
maybe 55500:50:07
oh how much we already have 55 divided by00:50:11
three 2 no it's00:50:15
darkness for you00:50:17
185 if I haven't forgotten how to count yet00:50:22
185 on the torus, it doesn’t matter what is important00:50:27
what is divisible by 3 without a remainder by me00:50:30
then I can make this comparison00:50:34
to the degree00:50:35
185 a00:50:47
4 to the 3rd power 185 you are 108 out of 500:50:53
again on the assumption that I can00:50:55
count it as 55500:50:59
that means 4 to the power of 555 is comparable from00:51:02
units of degree 185 or comparable to00:51:06
what's the difference between one and one00:51:08
any degree then all units00:51:18
Why00:51:20
me and00:51:21
28 because I multiplied by here00:51:26
I can press four seven at least 1600:51:29
what can I do with seven, it doesn’t matter00:51:33
you can't compare the number 3 cubed00:51:35
module 3 in two ways00:51:37
no indicator cannot be changed00:51:40
let's not change the indicators guys one more time00:51:42
we change only the numbers show the numbers themselves00:51:46
no need to have00:51:48
that is00:51:50
here I can’t replace 555 with00:51:53
zero when counting the remainder in division00:51:56
for a three I can only get the number 13 itself00:51:58
change00:51:59
that's so clear 5 it was done by a hundred for00:52:04
the tricks were shown now but it would be nice00:52:07
so that you understand this method too00:52:10
yes, that is, as it were00:52:13
just when I talk about comparison00:52:15
module I can standard this is this00:52:17
a phrase explaining why the leftovers00:52:19
get stuck replace with two three00:52:22
lines00:52:23
not so gypsy, that is, quite00:52:26
clear focus and here again yes guys00:52:30
Who00:52:32
This is the first time I've actually encountered this00:52:35
take some number there for example00:52:38
five and00:52:40
count, I don’t know, there are five00:52:44
one hundred and first degree modulo 700:52:49
look how00:52:52
leftovers work00:52:56
when will there be a period of looping like this00:52:58
further we are talking about this today in fact00:53:00
we'll talk more, we'll come back to this00:53:02
now this is too detailed00:53:05
the topic will not be revealed soon we will all get to it00:53:07
it will come just as long as you look at how00:53:10
comparison by sea works00:53:13
So00:53:14
let's move on00:53:19
we clean everything up00:53:22
I hope I haven't killed anyone yet00:53:24
there was no such goal in the first module00:53:32
leftovers00:53:35
Somehow it can be depicted geometrically00:53:39
For what00:53:41
why just why do you need this if we00:53:46
very big or do we just know00:53:49
that it is divisible by three, how to prove00:53:51
what then also the remainder 1 also divide it00:53:54
by 300:53:57
no, but00:54:00
now you mean if we had00:54:04
there was also a number of 555 Dacians, a larger number00:54:07
which is divisible by three we are in00:54:10
in principle we understand this to not be given00:54:12
I want to, yes, that is, relatively speaking, if00:54:15
we solved the previous problem for the number00:54:18
4 to the power 1 2 3 4 5 6 7 8 900:54:25
like yes, the number is divisible by three00:54:28
the sum of the numbers, but I don’t want to00:54:30
highlighting the problem can be solved very simply00:54:33
let's have this number designated as tights and00:54:37
let's understand that this is the same as 4u by00:54:41
tightly00:54:43
that degrees are multiplied00:54:46
I would have to do this from 15500:54:50
they called me a very lazy person00:54:54
so here it is00:54:58
So00:55:01
good morning Yaroslav00:55:05
Well, at least the functions or the screen00:55:09
of course yes very soon it will be00:55:11
literally one more two exercises in that00:55:15
same topic, look, the third task is very00:55:18
cool00:55:19
often in time she is just the third00:55:22
Let's sort it out and we'll get there gradually00:55:24
go to Ferm's little theorem and00:55:28
so a minus 2b00:55:32
are divisible by m00:55:37
& m minus 3 d are divided by m00:55:42
prove that from00:55:46
-6 b d00:55:49
divided by m00:55:54
this is already serious this is already serious00:56:01
A if a is divisible by b it is not necessary00:56:05
must have the same balances00:56:07
for some people, of course, there is no module00:56:08
of course not necessarily of course not00:56:10
definitely here00:56:13
let's start and look at the whole trick here00:56:16
that if we try now00:56:19
solve this problem honestly without hands00:56:24
modulo comparisons, what can we do?00:56:25
we can add we can multiply here00:56:28
imagine that we would multiply to00:56:30
so that it turns out father minus 2 b00:56:34
minus 3 a b plus 6 b.d.00:56:38
girl on m00:56:41
but has it become easier for us and there is00:56:45
father and 6 b d well first of all for some reason00:56:48
there is a plus sign in the second one here00:56:51
some negative terms which00:56:53
I'd rather throw them out altogether00:56:56
it’s unclear what to do next and if00:56:58
work with tests the world shared by suits but00:57:02
if we move on to comparisons by00:57:05
nothing will be left for the module from the task00:57:07
let's rewrite the condition differently00:57:09
if a minus 2b is divisible by m then you can00:57:13
write that a is comparable to 2b modulo m00:57:19
if c minus 3d diy is the price mtc comparable00:57:23
with 3d modulo m00:57:25
everything's great then let's do it00:57:28
Let's write that the product is multiplied by00:57:31
left to left right to right00:57:34
father is comparable to 6 b d modulo m a from00:57:39
it already obviously follows that they00:57:41
the difference is simply by definition of comparisons00:57:44
are divided into00:57:51
just a property of comparisons one more time00:57:54
besides, without comparison such a task00:57:56
it would be extremely unpleasant to decide00:57:59
as unpleasant as possible00:58:03
so now is the chance to ask a question00:58:08
and because gradually we will00:58:10
go to the little theorem truss i00:58:12
Now I’ll tell you some kind of theory00:58:15
pretty cool00:58:17
How is it possible for Ksenia to have something like this?00:58:20
probably can't but you just know00:58:24
on some00:58:25
qualifying stage like there, in principle, yes00:58:29
there find the remainder 4 to the power of five hundred00:58:31
fifty-five definitions for 9 to00:58:33
no, but as part of the task, of course yes00:58:38
Of course, this could easily not be from00:58:42
I don’t have a scientific degree, I’m kind of in00:58:44
teaching00:58:46
I teach00:58:48
so for some scientific work00:58:50
of course there is no time left00:58:52
about the fields no, we don’t need them now00:58:59
flying00:59:01
Greek application of number theory00:59:03
Well, right off the bat I don’t remember the name and00:59:06
I can’t google it, you can’t wait00:59:08
break is not there to brighten up remind your00:59:11
in the question there is something about some00:59:14
what are integer lattices?00:59:18
then I'll take a sip on Piglet and we'll00:59:22
let's continue with a small loss of the company00:59:47
don't go let me decide when00:59:49
speed up but I told the first block he00:59:52
actually it will be quite detailed00:59:54
so that we don't kill anyone anymore00:59:59
will01:00:00
faster01:00:02
So01:00:04
Yes, can I congratulate you, you survived01:00:07
one hour01:00:10
me and u01:00:18
mother is playing around and so we'll talk01:00:21
I'll tell you about Ferm's little theorem01:00:23
quite a lot of theories01:00:25
which now which may not be01:00:28
written on pieces of paper because there is no01:00:30
I was going to write everything at once but first01:00:33
first in short I can please you now01:00:37
you look yes01:00:39
in five days01:00:41
4501:00:43
under01:00:46
video in the description there will be a link to01:00:49
notes and01:00:51
so we'll try there accordingly01:00:53
display but in general more or less everything01:00:56
what was said in writing01:00:57
so that they can repeat if anything and if01:01:00
you seem to01:01:02
you're going to figure it out, relatively speaking01:01:04
Well, they didn’t know anything, I’m just out of control01:01:07
then it will be useful to repeat there as01:01:09
at least read the synopsis here01:01:12
So01:01:16
So01:01:17
from afar I will start talking about01:01:22
here is the set of all integers01:01:27
in general the whole honestly is 0 1 2 3 and so on01:01:30
then minus 1 minus 2 minus 3 minus 4 and01:01:36
so on, these are all integers01:01:42
yes, of course, as long as you are new of course01:01:45
but I don’t know how you can not like it01:01:48
so beautiful01:01:50
streamer and as it were, but have a conscience01:01:53
Please01:01:56
So01:02:01
this is what integers look like and now01:02:04
recipe for yourself what I was talking about01:02:07
module m 0 modulo five a01:02:11
Module 5 agree that integers01:02:14
it doesn’t look right at all, that is a minus01:02:17
three is the same as two01:02:20
Well, there's no difference between them at all01:02:22
there01:02:23
they compare each other modulo five01:02:26
three is the same as minus two01:02:28
minus four is the same as one and01:02:33
actually modulo emma sour01:02:36
look like this01:02:38
I love these bags draw a bag01:02:42
on which is written a zero pouch on01:02:45
which says one bag on01:02:49
which says two and so on01:02:52
a bag with m minus 1 written on it and01:02:56
yes it's a bag01:02:58
not some potato with a mustache01:03:02
012 and so on m minus 101:03:06
in fact these are not just numbers01:03:10
I took this and01:03:17
stuffed all the integers into little bags01:03:24
took it and painted it, shoved it into bags, well01:03:28
for example, where is the number minus 1?01:03:30
minus01:03:31
I send one unit to this bag01:03:35
where did the number 3 m plus one end up?01:03:40
she ended up in this bag and so01:03:44
further, that is, we stuffed all the numbers into01:03:47
bags, well, it’s clear why01:03:51
the thing is that now everything is clear from01:03:54
I can perceive one bag01:03:56
the same but what difference does it make to me?01:03:59
I'll take it out of the bag 2 is it worthy from there?01:04:02
can I get a deuce there m + 2 or 201:04:06
minus m it doesn’t matter all these numbers will be01:04:09
give the same remainder two01:04:11
definitions on01:04:13
I hope it's clear yes01:04:15
here it is01:04:17
now let's understand what this is for01:04:21
it's already done in the chat01:04:23
spoilers have started, full system of deductions01:04:26
yes, let's introduce such a concept as01:04:29
psv01:04:32
complete school system01:04:38
deduction system01:04:45
so yeah01:04:47
what's on the fingers?01:04:51
it works like this01:04:56
let's just from each bag01:04:58
take any number one at a time and01:05:01
let's call it a complete system01:05:03
deductions01:05:07
noir doesn't sound so pretentious, but it's not01:05:09
very similar to science they took the number from01:05:12
each bag so let us01:05:14
let's try to understand what properties it has01:05:16
this beautiful object why is it01:05:17
generally needed01:05:19
so let's go into a little more detail01:05:24
then let's describe this set01:05:28
cleaned from m01:05:32
important point, complete deductions01:05:35
let me write here modulo m01:05:38
Of course, it is not a complete system of deductions01:05:41
goes by itself it goes with01:05:43
the module I’m talking about in general01:05:45
that is, now let's agree01:05:47
that for the next half hour or hour we will all01:05:52
time we are talking about the same module m01:05:56
do not change horses at the crossing module m01:06:02
yes yes yes yes yes yes yes but attitude gives way01:06:05
to the valence of exactly these words I wanted01:06:07
to avoid01:06:08
set of m numbers01:06:12
giving01:06:15
all possible01:06:21
leftovers01:06:23
modulo m01:06:30
but not p&g, you wanted something01:06:33
new who complained01:06:36
It’s better there the same thing will happen with us now01:06:42
module m01:06:43
learn to see the difference there really isn't01:06:46
because now I’m not only talking about the little one01:06:48
I'm going to talk about the farm theorem first01:06:51
there will be few that the company will want later01:06:52
generalize for the derivative then01:06:54
arbitrary module01:06:57
here it is01:06:59
Well, let's understand what properties o01:07:03
a complete system of deductions is generally clear01:07:07
what if I take m numbers and they have01:07:11
there will be various leftovers give this too01:07:14
I'll write it differently01:07:20
m numbers01:07:23
giving different modulo remainders01:07:29
different rates01:07:31
modulo01:07:37
Here01:07:41
let's understand why it's the same thing01:07:43
why in this case it doesn’t matter to me if I take it01:07:46
am I01:07:48
all possible leftovers or I take01:07:53
various leftovers the thing is that I took01:07:56
exactly m numbers I'm generally emo01:07:59
various residues modulo m01:08:02
here they are from left to right from zero to minus 101:08:09
it seems clear that if I took everything01:08:12
possible means I owe everyone01:08:15
bag, select exactly one number and01:08:17
they will be different01:08:19
dial from I took various and their m pieces01:08:22
that means I took no more from each bear01:08:25
than one number and to dial exactly m01:08:28
from each number you need to take something01:08:31
it is clear that these are the same thing01:08:33
Here01:08:36
Well, why are we talking about this now?01:08:39
is that if suddenly it happens that we01:08:42
we will know about numbers only one of01:08:44
these properties01:08:46
for example, the fact that they are different01:08:49
we will conclude that this is complete01:08:51
deduction system01:08:54
well, I hope it’s not clear yet01:08:59
I try to do it all on my fingers more or less01:09:03
tell no relation there01:09:06
equivalence, what the hell with such words01:09:08
Means01:09:10
Let's01:09:12
two properties complete system of deductions we01:09:15
we realize01:09:18
psv01:09:20
+ a equals psv01:09:24
what do I mean let me explain here01:09:28
I have numbers leftovers x1 x2 and so on01:09:33
further x m.i.01:09:38
I'll give everyone a resignation, otherwise I'll be done01:09:41
I'll get the full deduction system here01:09:43
what I mean01:09:53
these numbers are also a complete system you01:09:56
that in principle it’s obvious, well look01:10:00
there are still m of them and they are still giving01:10:04
different leftovers why let's do this01:10:07
formally prove01:10:09
if not and t plus a01:10:12
comparable to their je t + a modulo m well and01:10:17
comparison can subtract the same01:10:20
number yes then x will be comparable to ig01:10:23
g there modulo m01:10:25
no what happens I took it originally01:10:28
two numbers that are comparable to each other then01:10:30
emits the same remains there was no such thing01:10:34
that means this can’t happen either01:10:38
so when we complete the deduction system01:10:41
add the same number then01:10:42
a complete system of deductions is obtained01:10:47
but this is not a very useful property01:10:50
wrote it out just like that01:10:55
more useful property this is what01:10:57
psv01:11:00
multiply by and equals01:11:03
psv01:11:05
If01:11:08
a&m are relatively prime01:11:16
if and are relatively prime01:11:20
then when multiplying the complete system of residues01:11:24
on and that is, each number on the base01:11:26
it turns out the complete system of deductions is here01:11:29
this is actually more complicated01:11:31
properties01:11:40
before01:11:42
of course yes to everyone01:11:47
that look at the evidence itself01:11:49
in fact the idea is exactly the same as here01:11:52
I had a complete system of deductions to x1 x201:11:56
and so on xm01:11:58
what will she turn into01:12:01
when multiplied by a it turns into x 101:12:05
multiply by 2 multiply by a and so on01:12:08
x m multiplied by a01:12:15
00 is generally a generally accepted designation01:12:18
lower 3 letters down01:12:21
just write in round brackets here01:12:26
that's a but01:12:29
no I didn't miss it again01:12:33
normal and generally accepted designation01:12:35
just yes, as if the school usually says not01:12:40
Let's all go back to school level01:12:42
write not a01:12:46
not a and I love the emoticon wonderful01:12:49
youtube like this01:12:52
would have been a long time ago without leaving then it would be good01:12:54
happened01:12:56
So01:12:59
Well, look, let's understand, let me01:13:01
I ask you what property is this01:13:04
new number system01:13:06
from photo01:13:08
this is the property of these numbers I can01:13:12
prove to be sure that01:13:14
it turned out to be a complete system of deductions01:13:30
what is this what01:13:31
mutual simplicity01:13:33
for some reason it’s very mutually simple01:13:37
no, and again about what in multiplication01:13:40
now guys look again01:13:44
there was a complete system of deductions I multiplied by01:13:48
but I got these numbers, it’s not about them01:13:53
enough proof to draw a conclusion01:13:56
that this is again a complete system of deductions01:14:00
that's what these are different leftovers01:14:04
there are different remains of them01:14:06
now yes01:14:08
it means Mattie if I prove that these01:14:11
numbers different remainders modulo m01:14:14
then I will get a set of m numbers giving01:14:18
different remainders modulo m, well, sort of01:14:22
agree01:14:23
then this is obviously a complete system01:14:26
deductions, we're just talking about this with you01:14:29
they said that this property is enough01:14:32
in order to conclude that everything01:14:33
there will be leftovers but there will be m of them01:14:36
that means everything is ok01:14:40
what are the topics at the marathon? all of which01:14:44
useful for olympiads guys who are on our01:14:47
marathon they know that we have already started everything01:14:50
study and by the way, well, how much will there be01:14:52
restart but now, well, better now01:14:55
obviously won't join for a month01:14:58
skip it in a month or so01:15:00
approximately until there will be a restart01:15:03
I’ll say something else about this, by the way, there’s more if01:15:05
look at the description there it's skin 1001:15:08
here's the discount for us01:15:15
yes yes yes yes yes yes yes here are all kinds01:15:18
I wanted to talk about the group again01:15:19
I would like to avoid the flow, after all, I'm in the end01:15:22
After all, I’m now communicating with schoolchildren in01:15:25
first of all so01:15:28
thanks guys thanks01:15:34
understood why it is generally enough to prove01:15:37
that these numbers give different remainders01:15:39
module m if this is so then there is still m01:15:42
numbers they give different remainders that means everything01:15:45
there will be leftovers there, which means it’s complete01:15:48
system of deductions that is really from01:15:49
each bag one leftover01:15:54
Here01:15:57
okay let's do it something like this01:16:01
we did it now01:16:03
let x & t multiplied by a is comparable to ig01:16:08
g there multiplied by a modulo m that is01:16:12
let's assume the opposite that we have some01:16:15
two01:16:16
the numbers below are the same remainders and01:16:20
that once again Arsenia hasn’t even started yet01:16:24
prove01:16:25
plus I’ll remind you that you can always01:16:27
rewind and fast forward to watch01:16:29
at some point I just remind you of01:16:32
just in case01:16:34
so look I get this01:16:37
property is correct01:16:43
A01:16:44
Well, of course, if they are comparable, let's do it01:16:46
we rewrite this by definition if01:16:49
comparable modulo m means the difference and01:16:51
some and multiply by a minus01:16:54
x je t multiplied by a divided by m01:17:01
man olympiad01:17:07
so let's take it out of the equation01:17:23
because of this we got to the hotel01:17:28
but look I had a condition that but01:17:32
numbers and m ralph units that is, but01:17:35
This doesn't help at all for divisibility by01:17:38
there are no common factors here01:17:41
This means that this number is entirely divisible by01:17:44
on m01:17:46
this is the conclusion I want to draw:01:17:49
the difference between the numbers is divisible by m, so x is01:17:52
comparable to their Jews modulo m01:17:57
yes there will be a recording01:17:59
well, how can xetai be comparable to01:18:02
ig g there modulo m01:18:06
but this would not be a complete system of deductions01:18:09
if so01:18:12
I compare two numbers modulo m like this01:18:15
it can not be01:18:16
they can even say there will be a recording but01:18:18
only if you like and share01:18:22
sorry excuse me so so so so01:18:25
I work for your likes01:18:28
why are they divisible by m they just don't01:18:31
are divided into m & m mutually prime to me01:18:34
it is not enough to check that they are shared01:18:36
on m realize 2 by 3 is perfectly divisible by01:18:40
six although not deuce no video for 6 not01:18:42
shares I check mutual just stupid01:18:45
according to the condition, the numbers a m are relatively prime, that is,01:18:48
they have no common factors01:18:52
figured it out01:18:58
there is something that you and I have now proved01:19:02
proved that you can multiply the total01:19:05
system on a number is coprime with01:19:07
module and nothing changes01:19:11
well if everything is still completely deductible01:19:13
it turns out01:19:18
cosmo it is01:19:22
Kazakh or whatever the indicator is01:19:26
well what can happen when we get to01:19:28
indicators even we will if you throw it off01:19:31
let's try to solve01:19:33
Here01:19:38
Fine01:19:40
So01:19:42
I didn’t kill anyone guys, that’s what it is now01:19:44
it's important to understand what's going on01:19:47
firstly this is for more than01:19:51
half of it seems to me to be still new01:19:53
all sorts of topics these are given here01:19:55
the complete system of deductions is now one more01:19:58
there will be a reduced system of deductions here01:20:01
so guys, let's talk about dislikes01:20:04
forgot01:20:06
not interested01:20:10
that is01:20:12
psv full system of deductions that's all01:20:17
don't reconsider this moment then01:20:19
let's keep it that way01:20:24
guys you write dripped like really01:20:28
something is unclear or01:20:31
so if how would it be01:20:37
sarcasm please don't forget01:20:39
putting Malik the clown streamer is stupid and01:20:42
doesn't understand sarcasm01:20:47
so far so good and great01:20:50
Amazing01:20:52
what about guys if it starts01:20:55
it’s a little hard, well, first of all, yes, I would01:20:57
I remind you just in case what the hell this is01:21:00
Olympics, that is, I can’t be here01:21:02
repeat 5 times while the rest01:21:05
guys are waiting for something interesting if01:21:09
something is unclear, really reconsider yes01:21:11
there is x2 acceleration if it’s not clear01:21:14
I can ask you a specific question01:21:16
I’ll answer, it works for no reason01:21:19
I don’t want to repeat it a bunch of times, okay01:21:23
so now we come to the little three firm01:21:27
and there we are01:21:31
it will be proven with the help of PSV and01:21:34
more precisely PSV01:21:37
here it is01:21:39
here some question asks what01:21:42
I didn't understand the question ok01:21:46
then look, thank you, you figured it out01:21:49
there is a complete system of deductions01:21:51
the main thing we must remember01:21:53
yes, nothing is really needed yet01:21:56
we just needed her for this01:21:58
so that now it’s easier to sing01:22:00
another concept is reduced system01:22:02
deductions01:22:10
the system of deductions is given01:22:16
P01:22:17
sv01:22:20
reduced system01:22:33
what is the difference from the full topic of deductions we01:22:37
just took and removed the remains that were not01:22:41
are coprime to the modulus, that is01:22:43
left only those who mutually01:22:45
easy with mods01:22:50
to the Pension Fund01:22:55
left only01:22:58
bags01:23:04
for brother and with module01:23:20
to begin with, the problem here is that01:23:23
V01:23:25
their psv start with the same01:23:28
so I will write the letter psv01:23:30
this is a fairly common designation01:23:33
at least there's a bunch there01:23:36
literature you can find all this not me01:23:40
came up with the name slicer, it’s not like01:23:43
taken from the ceiling01:23:45
then look at01:23:53
problem in but for arbitrary module m01:23:57
we are actually really bad with you01:23:59
we understand what will happen, what purely01:24:03
will disappear which will remain therefore it will be01:24:06
it is useful to first highlight a separate01:24:08
the case when the modulus is a prime number01:24:14
let further01:24:19
m equals p is the number01:24:33
bags yes this is where we are01:24:36
signed for the remainder01:24:39
such bags will remain if it were01:24:41
prime number for reduced system01:24:44
deductions, well, of course the zero will be removed01:24:48
there will only be one two left and so on01:24:53
further the bag on and cons01:24:57
that is, only the remainder 0 drops 001:25:01
not all drop out remain 0 drop out but01:25:06
everyone else remains01:25:11
let's think about the properties01:25:14
still persist can we01:25:16
add an arbitrary number a and01:25:19
hope that what remains is given01:25:21
the deduction system is a question that we are with you01:25:24
let's discuss it verbally now01:25:31
Pavel definitely knows something01:25:34
you can't add it, obviously yes01:25:37
imagine for example modulo five01:25:38
yes there was one-two-three-four you01:25:41
added two to everything and got two01:25:43
leftovers 3 4 5 6 but damn it's a five01:25:49
it sucks, that is, it’s clear that01:25:52
adding is bad because most likely01:25:55
you will get a zero among the leftovers later01:25:58
we won't add more01:26:00
The funny thing is that you can still multiply01:26:05
I'll remove this part01:26:08
adds we have forgotten how to01:26:11
you can still multiply01:26:15
they, well, let me clean it up too, at least01:26:18
there was a full system of deductions now there will be01:26:20
completely different proof01:26:21
so now I'll erase everything and write again01:26:25
times the same thing only for01:26:26
for the reduced system of deductions, see01:26:29
Here01:26:31
it was PSV01:26:34
if I multiply psv by and then too01:26:38
it turns out psv with the same condition01:26:41
a&m01:26:43
in our case p01:26:47
relatively prime01:26:49
I’ll write but so as not to confuse anyone01:26:55
Here01:26:57
A01:27:01
then it started01:27:03
this one is offended01:27:05
We don’t discuss algebra, everything is clear, clear01:27:11
so that means let’s top up the block with PSV right now01:27:16
there is little room for everyone in companies like this01:27:18
Let's take a 5 minute break before01:27:22
they will not always be exactly accurate01:27:25
schedule but approximately I will try01:27:27
so that it doesn't drip01:27:29
here psv multiply by this again prs01:27:35
well look, then let's do it anyway01:27:40
Let's discuss first what we need01:27:42
know about set of numbers01:27:47
to claim that this is given01:27:50
deduction system when we talk about01:27:52
prime number p well first of all we need01:27:55
know that the numbers are by and -101:28:02
auto need to know what01:28:05
numbers are coprime to modulus01:28:12
in on in case it is simply not divisible by p but01:28:16
in fact it's better to speak right away01:28:17
the right words that will come later01:28:19
convenient to transfer to any location01:28:24
third01:28:29
the numbers give01:28:33
different a-stars01:28:37
or01:28:40
isla give and01:28:43
mutually prime cars01:28:50
let's realize that again we01:28:53
it is enough to know any of these properties01:28:56
we don't need to prove both01:28:58
if I know that the numbers give different01:29:01
their remains are each and -1 piece, they are all mutual01:29:03
simple v.p. well that means 0 is not there at all01:29:06
the rest should be01:29:09
obviously on the contrary, I know what they give01:29:11
all mutual simple remainders of them by and -101:29:13
thing there can't be the same01:29:15
because it just doesn’t mean anything01:29:17
it won't be enough01:29:19
the recording will remain of course01:29:23
thief and exactly where I will be in the world now01:29:26
proves this property I really am01:29:28
I’ll check what different people give purely01:29:31
leftovers I won't check what's there01:29:34
they will do everything possible to check this01:29:36
it’s convenient to check that they give different01:29:38
the rest is very easy in the second it is written01:29:41
numbers are coprime r.p. that is, purely01:29:44
in the given system of deductions it is still01:29:46
must follow the tracks on her and be01:29:48
mutually simple St. Petersburg this is us too now01:29:50
I'll check, so let's do it again01:29:54
numbers x1 x2 and so on x with index p01:29:58
minus 101:29:59
agree there are already n minus 1 pieces01:30:04
we multiply them by and it turns out ex1 ex2 and01:30:09
so on and01:30:11
x with index p minus 101:30:20
[music]01:30:23
why do we remove only 0 and because01:30:27
modulus is now a prime number one more time we01:30:31
Now we are considering only the case when01:30:33
fashion prime number because now I01:30:35
I want to move towards Fermor Elvira01:30:39
then we'll take a look while you're a little company01:30:48
so that means, well, let's see what01:30:51
properties of this set of numbers immediately01:30:52
it is obvious that the numbers are -101:30:55
thing ticked01:30:58
what is purely still mutually right prefers01:31:02
but look01:31:04
Anapa didn't share01:31:07
x that’s me too Anapa didn’t share that’s why01:31:11
it is obvious that a x will not be divided01:31:13
on p01:31:15
still01:31:22
start checked01:31:26
why do the numbers give different resignation again01:31:30
let's check with our hands let a and01:31:32
and x is comparable to xdd mod n01:31:39
well, you know what needs to be done01:31:41
look at the differences everything is the same01:31:45
the difference x is minus their living is divided01:31:49
to p then01:31:51
but for me they are divided into n here yours is not01:31:54
only what is not equal because it is not01:31:56
is divisible by p then it means that it should01:31:58
divisible by p but then initially in01:32:01
given the system of deductions I had01:32:03
two identical remainders modulo n but not01:32:07
this could have happened before the contradiction01:32:09
happened01:32:12
Here01:32:17
wonderful, that is, we are still only01:32:20
knew with you that the one brought remains01:32:23
deduction system is correct01:32:26
give so who understood what happened01:32:31
reduced system minus that we01:32:32
proved that when multiplied by a still01:32:35
the reduced system of deductions remains01:32:37
put a plus sign on chat01:32:43
But01:32:45
in class it’s basically better then first01:32:47
you can see01:32:49
wrap it up but it’s quite possible to understand everything01:32:53
there are sixth graders here01:32:56
I know for sure01:32:59
marcin but you repeat it then I'm here01:33:05
just asked for a bunch of pluses01:33:06
please write again01:33:08
just once01:33:10
about your own when taken according to simple or01:33:13
you can simply throw away the one that doesn’t fit01:33:16
conditions for difficult well01:33:19
pps yours is not always taken in simple terms01:33:22
there is I first determined the peerage then01:33:24
said that the modulus is a prime number01:33:26
it didn't just happen in this way01:33:29
okay, first we define what01:33:31
this is what we're dealing with now01:33:34
for a prime number01:33:40
well, the maximum balls are not fully open01:33:45
screen otherwise I wouldn't have seen it01:33:47
I read what I see if something important is not01:33:50
hesitate to repeat your question, how about it?01:33:53
would01:33:54
show shui01:33:56
if not your own01:33:58
ours you are there01:34:02
excellent and I think even some01:34:06
Bye01:34:07
figure it out how01:34:10
Well, actually I’ll get it, I promised01:34:14
prove Ferm's little theorem we are almost01:34:16
ready we're almost ready left01:34:18
literally one obvious property01:34:21
here's a trailer for myself, I took one given01:34:26
system of deductions and01:34:35
another01:34:42
I'll multiply the numbers inside here01:34:48
multiplied and01:34:50
multiplied inside here01:34:56
I claim that the resulting two numbers01:34:59
comparable in modulus and but helped p01:35:05
that is, it doesn’t matter to us which one we took01:35:07
reduced system of deductions if we01:35:09
multiply all the remainders inside01:35:11
of the reduced system of deductions we get01:35:13
same meaning01:35:16
well the obvious thing is that01:35:19
the given system of deductions contains01:35:21
numbers with equal remainders01:35:25
so what's the difference01:35:27
like will I multiply five or minus01:35:31
two if I'm talking about module 701:35:34
therefore this is an obvious property of our01:35:36
modulo comparisons01:35:38
Yes01:35:40
Yes, the tenth grade can and needs this01:35:43
look01:35:46
of course, especially when they start01:35:50
an even more complicated thing01:35:53
yes of course psv one by one it is possible01:35:55
module yes we talked about this half an hour ago01:35:59
They said yes, there will be a module soon01:36:02
fixed01:36:04
it’s clear the properties and what’s involved01:36:07
multiplication results in the same thing01:36:09
now let's do this trick01:36:12
let's take the reduced system of deductions01:36:15
the simplest01:36:18
day01:36:23
1 2 3 and so on in minuses01:36:30
the simplest reduced system01:36:32
deductions were simply taken one balance at a time01:36:35
multiplied by a and got a01:36:42
two and three and so on by and -1 by a01:36:53
Same01:36:55
deduction system01:36:57
proved this property that the sentence on01:37:00
and the reduced system of deductions remains01:37:02
the given deduction scheme01:37:04
multiply numbers here multiply01:37:06
numbers here01:37:08
what happens here is one in 20 and so on01:37:11
further by -101:37:16
multiply the numbers there01:37:20
she 2 a sodium and so on for p minus 101:37:25
on a01:37:29
they are comparable modulo n well because01:37:32
leave here anyway from 1 to n minus01:37:35
1 just in some other order01:37:37
we multiply them and it will still work out01:37:40
the same thing, we're just talking about this01:37:41
they said01:37:45
yeah01:37:46
look but let me transform it a little01:37:50
let's take from each factor from01:37:54
q I will take a to the power01:37:57
well, like two rsvp light and a set of numbers numbers01:38:03
so for example modulo five I can01:38:06
take one two three four I can take01:38:10
2468 and01:38:12
both then reduced system01:38:14
deductions but they look different before01:38:17
different numbers01:38:20
leftovers to leftovers they have of course01:38:22
the same01:38:28
that's how much endurance each aspo has01:38:32
it turns out01:38:34
powers p minus 101:38:39
well because I have this multiplication yes01:38:41
Some guys here are confused I think01:38:43
what you need is simple and to the first degree01:38:45
bear this multiplication with each bear01:38:47
hq so it turns out n minus 1 hk01:38:50
what's left is just one on two01:38:53
by three and so on by p minus 101:39:00
I think I said modulo five01:39:04
I may have made a mistake, but I was talking about the module01:39:07
501:39:11
[music]01:39:12
so look we got this01:39:15
here is a comparison there is a common factor01:39:17
p minus 1 factorial let's do it differently01:39:19
rewrite apple proved that p minus 101:39:22
factorial is comparable to a to the power of b minus01:39:25
1 by p minus 1 factorial01:39:29
modulo n01:39:32
give the difference we write a01:39:37
to the power of b minus 1 minus 1 times01:39:42
p minus 1 factorial01:39:45
divisible by p01:39:52
p minus one factorial is coprime with01:39:57
P01:39:59
because it's a prime number01:40:03
there is not a single number here01:40:05
divisible by p01:40:07
this means that this bracket is divisible by p01:40:13
this implies01:40:15
then a to the power of b minus 101:40:19
comparable to unity mod n and01:40:23
this is called Fermat's little theorem01:40:31
yes here with division as it were01:40:34
Now you and I actually understand01:40:38
what can be divided if the module and tones what01:40:40
we divide are mutually prime because then01:40:43
if I rewrite it this way and how01:40:46
difference and take out the common factor for01:40:49
brackets I can remove it, it doesn’t carry01:40:52
no01:40:53
semantic load but the sense of multipliers01:40:56
he carries simple ones for himself that are needed for01:40:59
divisibility01:41:00
that's it, but for now I'll write it down in detail01:41:04
in detail01:41:07
Here01:41:15
we have proven little more than a company01:41:18
why by and -1 factorial 2 is simply sp01:41:21
We are now considering the case when01:41:23
prime number therefore numbers from one to01:41:27
-1 is not divisible by p since it is prime01:41:30
this is enough to conclude that01:41:32
n minus 1 factorial is not divisible by p01:41:35
because the prime number is short01:41:38
answer a01:41:40
to the power of p minus 1 minus 1 see i01:41:45
wrote the difference and01:41:47
right away you are not from n minus 1 factorial well01:41:52
I just missed one line01:41:54
transformations on p minus 1 factorial01:41:56
can be taken out of brackets01:41:59
it turns out a to the power of minus 1 minus 101:42:02
hello him01:42:09
Well, these are the fundamental ones01:42:12
proof01:42:13
that is, she doesn't seem to look like01:42:16
some kind of trick if we talk about numbers01:42:20
Of course you need to know exactly him, otherwise01:42:23
what I told you at open days01:42:25
towers, well, it’s like a cool trick there01:42:29
and look how it happened01:42:33
Gregory, well, because look01:42:35
the product is divisible by p because this01:42:38
I subtracted the difference from the right side to the left01:42:42
by definition of modulo comparison01:42:46
the difference is divided by p then I just01:42:48
took out n minus 1 factorial after digging01:42:52
it turned out that the product of these01:42:54
two numbers divisible by p01:42:57
I realized that p minus 1 factorial to me01:43:00
there is no help here there are no01:43:02
common factors sp means exactly this01:43:05
this thing is divisible by p then they immediately love it01:43:08
please, if I answer you01:43:10
your question, how do you respond, write to01:43:12
chat did not understand01:43:14
so it will be somehow better for us01:43:17
to interact01:43:23
everything is great, still some questions01:43:29
in fact there is evidence using01:43:32
groups just in other words arthur01:43:36
in fact, I didn’t say either myself or01:43:38
once the words are a group but in essence it’s the same01:43:40
most01:43:42
first course and the phenomenon is understandable personally01:43:45
great, everything is clear01:43:49
super-super-super01:43:56
1 minus x to the power of minus 1 y works01:43:59
but what difference does it make how I write I can01:44:01
at least subtract the left from the right side at least01:44:03
and left and right Alexander, well, either01:44:05
specific question that I can answer01:44:07
answer or review then part01:44:09
proof in principle youtube01:44:11
allows you to do this01:44:16
a-ah powers p minus a divided by p and a01:44:20
to the power of minus 1 is comparable to one in01:44:21
Module n is the same thing01:44:24
that is, let's give it let's multiply it01:44:27
comparison on a we get a to the power of b01:44:29
comparable to modulo n01:44:36
but like Alexander I01:44:39
I can’t answer quickly I can’t repeat everything01:44:42
again01:44:43
why a01:44:46
degree is divisible at the limit because01:44:48
once again you understand where this line comes from01:44:52
took it just said it is divided differently01:44:55
on p01:44:57
in difference I would not put n minus 1 in parentheses01:45:00
factorial taken out01:45:06
the result is the product of two brackets01:45:09
means one of these brackets is divisible by01:45:11
p because p is a prime number01:45:13
this thing is not divided into p stages01:45:17
and -1 factorial n prime number and prime01:45:20
but not a single number that is there01:45:22
if it's on Monday it means this one01:45:25
parenthesis is divisible by p01:45:29
thank you number01:45:31
Well, that is, Artek will not understand one minus 101:45:33
divided by p means pharmacies not p minus 101:45:35
comparable to unity modulo01:45:39
w-why are the works of the PFP members01:45:42
comparable and because in fact it is01:45:44
we have the same numbers there, different balances01:45:48
but the set of residues there is the same both there and01:45:50
there are just all possible leftovers01:45:52
simply modulo n mutually prime spa01:45:55
And01:46:00
don't rush don't rush induction then it01:46:05
you and I will not go to the circle01:46:07
we are experiencing camilla yes yes yes yes yes yes01:46:16
no, but it seems clear that something01:46:20
it was possible not to realize for this it is possible01:46:23
will reconsider is not a problem, that is01:46:25
as if I’m saying right away if I was here right now01:46:28
complicated, it’s really better here are the first three01:46:31
look at it for an hour and then look at it there01:46:34
once again, rather than delving into complex topics at once01:46:39
here I'm really slow and01:46:43
even carefully explained in detail01:46:46
too slow I would have to01:46:48
a little bit of a climb to climb, well, lan we are01:46:51
we’ll do it without any problems, yes now it’s like01:46:54
break 5 minutes until 1150 5 thousand I still01:46:58
tell me now I won’t answer01:47:00
only to your questions on the topic or not01:47:03
until 11 5 you can relax with some tea01:47:09
make a moto fail01:47:13
why bank b k for what a01:47:18
in k it seemed that when I ask01:47:20
reschedule a lesson for a 12-hour stream01:47:23
then this is a suspicious mailing01:47:26
there was a picture like this01:47:29
type01:47:32
means new01:47:34
I’m poking the reason, here’s the message for me01:47:38
makes me like I’m writing hello to the student there01:47:42
Sunday I do a 12 hour stream01:47:45
I suggest rescheduling the class to Friday01:47:47
19 00 here we go01:47:50
that's what the ban is for, so the ban is for 12 hours01:47:54
spin01:47:56
before01:48:01
and two hours passed and generally for free01:48:03
I don't know I didn't notice I didn't feel01:48:08
after two hours of running I already felt01:48:10
I haven't noticed yet, it's too easy01:48:15
restart is planned in a month but at01:48:18
in fact I highly recommend01:48:20
I recommend that you connect now01:48:23
Moreover, many people will now have holidays01:48:25
let someone study in quarters, that is01:48:28
well, sort of01:48:29
naturally01:48:32
through to feel everything all all in01:48:35
Act Olegovich ok01:48:38
I've lost weight, maybe you can wear it under a T-shirt01:48:42
put on that doesn't squeeze my belly01:48:48
add of course everyone believed that01:48:51
you could send the link normally01:48:53
rickrolling but but01:48:55
I will eat but when you are a little big01:48:57
no breaks now good afternoon mr01:49:00
before this is so you guys for 4 more minutes01:49:02
breaker who wanted it there01:49:05
relax, get some fresh air01:49:09
do this before we continue01:49:21
eat01:50:40
I don't see Bayer on the stereo, not even a cult01:50:43
for a moment I was thinking about your direct post but there is01:50:47
one problem seems to me to be not very good01:50:49
popular, that is, many01:50:52
by default they are simply afraid of stereometry as01:50:55
even on our webinars, our course01:50:57
steamed ryu consistently half as many people01:51:00
watch than other webinars, here’s how01:51:03
would be on01:51:04
such a thing01:51:08
why exactly do I drink moto because in01:51:12
Unlike coffee, it doesn’t dry out my throat01:51:16
Alexander likes this01:51:19
shams don't need to spam some [ __ ] on01:51:24
then you can get the bank to talk about something01:51:27
at least write a set of letters01:51:30
the same stream on legal cambium or01:51:33
algebra for example maybe maybe01:51:36
May be01:51:40
why did they choose math olympiads01:51:42
very waist g well damn maybe because01:51:45
I've been a jury for seven years01:51:49
Olympics for example01:51:52
therefore durable until received the Olympics01:51:55
I'll choose math01:52:00
a miracle for students 12 hour stream01:52:03
highest from normal normal beautiful01:52:05
Beautiful01:52:07
beautiful 100 most important tasks of norms01:52:21
We’re not afraid and we don’t love them, well01:52:24
it's the same thing about us01:52:33
thank you diman thank you so it's time01:52:37
continued measures on my watch 155 like this01:52:41
what let's continue01:52:45
so we proved the small tower right next to01:52:48
Euler's theorem is next for us01:52:50
here it is01:52:52
heart if you understand everything now01:52:57
too bad if not because now I will01:53:01
in a sense doing the same thing but01:53:05
For01:53:06
any numbers not necessarily prime01:53:09
actually possible if something01:53:11
misunderstood even when we are now01:53:12
we'll do it again, it'll be better, but keep it simple01:53:16
hold on01:53:19
what is a and is an arbitrary number01:53:22
which is coprime with the module here01:53:25
this is an arbitrary number of a cat are we01:53:27
just watching01:53:29
So01:53:35
let's all move then01:53:39
forward a01:53:42
b and forward the game will come, then I probably01:53:46
I'll remove this part and01:53:51
now we will start talking about what01:53:55
is Euler's theorem and01:54:00
So01:54:02
A01:54:04
look, there you go01:54:09
the question arises what changes if m this01:54:13
not a prime number but a composite number here01:54:17
It’s important to realize that I really don’t ask you to01:54:19
Just01:54:23
Well01:54:26
listen and wow the magic is so real01:54:30
understand why we use a composite number01:54:33
we do it exactly the way we do it here01:54:37
what would be the problem if n weren't simple01:54:40
a compound number01:54:43
why would something change here who01:54:46
understands01:54:51
come the system never matches01:54:53
full arthur must for prime number 001:54:57
and besides, it was already removed01:55:00
let's go to the parameters and there Anna01:55:03
Vladimirov are starting very soon so01:55:05
okay guys, here are the Olympics nicknames01:55:08
you understand about the parameters, yes01:55:10
the first two hours there are most likely well01:55:12
there will definitely be a zero level, so01:55:17
it’s quite normal to watch this one01:55:20
stream up to 1400 then go to parameters01:55:23
especially if you highlight the cooking there01:55:25
to any physics and technology specialist, I think that Maxim01:55:28
Olegovich s doesn’t hear, he’ll support01:55:31
that's right guys write what if01:55:36
b is not a prime number when appears01:55:39
such a problem is when you01:55:42
here01:55:44
consider the difference you can no longer01:55:47
it's so easy to say about the metro01:55:50
will of course you can no longer take and01:55:53
say like a01:55:56
throw it out because it is not divisible by p01:56:00
can share with this thing if only01:56:03
you weren't a prime number, you would be common01:56:07
divisors and01:56:09
then I would not be able to conclude that01:56:12
this is divisible by p01:56:14
that's the difference, that's what it was when I bought it01:56:17
prime number01:56:18
respectively for the reduced system01:56:20
deductions I need to leave the numbers01:56:24
works that will be mutual01:56:26
easy with the module that's why up to01:56:28
definition I generally consider01:56:30
coprime numbers modulo01:56:33
so that later here01:56:36
it was possible for this common factor01:56:38
cut it down and so on01:56:42
a reasonable question arises, ok here it is01:56:44
we're going to keep it coprime01:56:46
numbers and how many are there?01:56:49
how coprime are the numbers?01:56:52
[music]01:57:00
some kind of smart things in principle, yes but01:57:02
but not everyone understands why they are now01:57:04
write well yes01:57:08
the problem is that no one knows everything01:57:11
no, they know, but it’s not like that’s a question01:57:15
which is easy to answer right away: how much01:57:17
numbers that will remain in the given01:57:21
deduction system, for example, let's01:57:23
we think we understand what if it happened01:57:26
prime number 5 then 4 numbers left01:57:31
modulo five we have 4 numbers prs in01:57:38
helped 6 who will tell when how many numbers01:57:42
psv01:57:46
before01:57:47
Alexander now we are up to her Crosby01:57:54
12 to is obtained modulo01:57:59
621 and 501:58:01
the remainder will remain 1 and 5 were the remainder01:58:05
zero one two three four five01:58:11
how not to cross out the zero01:58:15
mutually just six two three and01:58:18
four01:58:19
one and five remain01:58:24
two numbers01:58:28
2 numbers psv01:58:35
modulo 10 let's see for example01:58:37
well, simple, no longer interesting, we understand01:58:40
that in simple terms modules are just by and -101:58:42
all but 0 remain in my opinion 10 like this01:58:46
working was zero one two three four01:58:49
five six seven eight nine01:58:53
we must cross out the zero two01:58:55
four six eight a01:58:59
also an A01:59:02
that is, four numbers RSV01:59:13
Well, that is, let's be honest with everyone01:59:18
the law is not obvious according to which01:59:20
the number of numbers is determined01:59:26
1379 everything is correct you write 137901:59:30
nine remains too01:59:34
But01:59:36
so that we don't have to sell01:59:39
consider invented function or function01:59:43
Euler and01:59:47
from m01:59:58
yes of course you can decompose yes but now02:00:03
we won’t talk about this, here’s this for us02:00:05
right now it’s not very interesting because02:00:09
in principle we can do this02:00:11
perceive simply but abstractly well yes02:00:14
well, like, well, you can count, yes, even02:00:18
for a hundred you can basically count02:00:20
first crossed out all that are divisible02:00:21
by two then everything that is divisible by02:00:23
five for a prime number is generally02:00:25
-1 you can count, but you need to know02:00:28
decomposition of the number m is now a little02:00:30
not in our direction, yes, I almost promised02:00:35
do not touch upon the decomposition today02:00:37
the multipliers are almost02:00:41
well for n one more lot and laces02:00:45
means the function lira feat m is02:00:47
quantity02:00:50
numbers of natural numbers02:00:58
less than or equal to l and coprime02:01:02
prices02:01:06
this is how it is determined02:01:09
or02:01:22
our business02:01:25
It's basically written here that it's simple02:01:28
number of psp numbers02:01:31
in fact02:01:34
number of numbers rst02:01:40
why well because natural numbers02:01:44
from one to m give all possible02:01:46
remainders modulo m02:01:49
you just need to replace the m with a zero and oh well02:01:53
I’m just crossing out everything that’s not mutual02:01:55
simple02:01:57
leaving only mutually prime i02:02:00
I determine what this 5 means with psv02:02:02
5 m numbers left02:02:05
not strict by definition by definition02:02:09
so that the zero is also taken into account in reality02:02:11
almost always of course it is possible and strict02:02:13
put one on fiat, it's one02:02:17
that's how it should be, that's how it should be02:02:21
denis but let me decide how to write with02:02:25
with the ending of the word or without the ending yes I02:02:28
I'm not hosting the webinar02:02:33
I don't want this time02:02:38
so we figured it out02:02:42
that is, they simply introduced such a function feat02:02:45
m denoting the number of numbers not set02:02:48
numbers, namely the quantity02:02:52
ok then02:02:57
let's think about what properties02:03:00
the given deduction systems are preserved02:03:03
for an arbitrary module m02:03:10
well, yes, the same ones are saved, well02:03:15
look, let's be careful again02:03:17
one time before we prove this property for02:03:19
arbitrary module m we understand that we02:03:22
let's do the same now02:03:24
so I took psv well that is x1 x2 and so on02:03:30
further x by index and from m02:03:37
we understand that such an index is what it is02:03:39
their number of feta numbers and02:03:41
multiply by a02:03:44
I get ex1 ex2 and so on as index02:03:50
multiply by x with fiat index and02:03:57
what do I need to check what I cleaned well02:04:01
no longer n minus 1 and from m here and02:04:05
I can handle this fiat02:04:07
well, how could there still be so many of them left02:04:12
obviously02:04:14
under each multiplied the amount from this02:04:17
hasn't changed it's still obvious02:04:19
why are numbers still relatively prime to02:04:23
module m-va let me realize this02:04:27
is an arbitrary number a multiplied by x02:04:29
and d02:04:31
this number and m are mutually prime02:04:33
definition02:04:34
otherwise I wouldn’t have included him02:04:37
deduction system02:04:39
a&m are coprime because not is equal02:04:43
unit by condition02:04:46
here are these two numbers, well, this is the number 702:04:49
the price is simple this is mutually simple02:04:51
this means the product is also common02:04:53
Sam has no multipliers02:04:57
still mutually just with02:05:07
Well, it's probably worth a look first02:05:10
Yes02:05:19
So02:05:21
Well, the third property is that all numbers02:05:24
they also give various leftovers, well again02:05:27
let's write ok let it be kHz02:05:30
multiply by and comparability runs and02:05:33
multiply by a modulo m let's do it again02:05:37
Let us consider the difference a02:05:40
multiplied by x is minus their g d02:05:45
divided by ma02:05:48
m mutually prime means then x is02:05:52
minus x is divided by m and we are again02:05:55
came to the same problem but damn02:05:56
initially we didn’t have two numbers02:05:58
which give the same remainders means02:06:01
this can't happen02:06:02
so everything is still great02:06:05
this is the property that works02:06:08
multiplying the reduced system of residues by02:06:10
and we get the reduced system of residues02:06:13
it was preserved02:06:16
we will now prove the theorem02:06:18
Euler02:06:19
but I won’t formulate it yet, that is02:06:22
just trying to generalize for numbers02:06:26
difficult02:06:29
Here02:06:31
so it’s true that it’s obvious guys02:06:35
let me just say02:06:38
By the way, I'll do a bunch02:06:43
maybe guys, if there’s something unclear there02:06:46
then you ask immediately they are at the end02:06:50
so good good02:06:53
Well02:06:55
let's now02:06:58
let's do the same and just multiply02:07:01
number of reduced systems of deductions02:07:07
I think it's still obvious that02:07:10
if I take the 2 given systems02:07:12
deductions02:07:17
transfer if here02:07:21
multiplied02:07:23
multiplied here02:07:26
then we get the same thing modulo m02:07:30
because again the set of remainders is not02:07:32
changed02:07:40
Nuha02:07:41
let's take some02:07:43
the given system of deductions here I go to02:07:46
unfortunately I can’t take specifically 1 2 302:07:48
and so on n minus 1 is no longer02:07:51
the reduced system of deductions is simple02:07:53
let's take some okay x102:07:56
x2 and so on ikt with index fe dm02:08:02
let's take it02:08:04
multiply by a02:08:06
we get ex1 ex2 and so on and x c02:08:12
fiat index02:08:18
Maxim, well, in general02:08:22
not very cultured if you didn't listen02:08:29
part then ask again a bunch of times02:08:33
in general, I'm now trying to generalize for02:08:35
composite number but a big request02:08:37
guys are new02:08:40
listen, listen02:08:42
yeah ok but I'm just answering this question02:08:46
already seemed to be responsible for what we just did02:08:50
like two minutes ago02:08:55
that's exactly why we multiply psv well02:08:59
to prove Euler's theorem02:09:00
that is, still like me02:09:02
I'm just following the proof02:09:06
Ferm's theorem and I want to understand what is changing02:09:09
what doesn’t change and trying to understand which02:09:12
I will prove the theorem thereby02:09:20
make out yes, that is, for now you agree02:09:23
changes are minimal02:09:26
I just write the index differently, like this we02:09:31
we understand that if we multiply these02:09:32
honest, these will turn out to be the same thing02:09:35
By02:09:37
module m of course02:09:59
Well, let's take the peace tonic and we'll take it too02:10:03
let's remove hq from the right side.02:10:07
is changing02:10:09
let me erase it already02:10:15
1 tm but I already talked about this02:10:19
Euler function02:10:22
function number of natural numbers not02:10:25
superior and we are mutually just with m02:10:28
yes from the number of numbers in the given02:10:30
system of deductions according to an arbitrary module02:10:32
m02:10:39
but after02:10:40
she you prove because great02:10:43
Fedot didn’t work, then the evidence is well02:10:46
that stopped working because02:10:49
playback02:10:50
n minus 1 factorial can now02:10:53
divisible by p I can't deduce02:10:55
what exactly is different is divisible by p?02:10:59
it works so we do it again02:11:01
no, I'm trying to understand what makes sense02:11:04
it is clear that someone will gradually02:11:07
it's like a claw02:11:09
Unfortunately02:11:12
inevitability02:11:14
inevitability I just want it to02:11:16
this is but first of all we got by02:11:19
minimal losses secondly, well, like02:11:22
those who listened really understood02:11:26
because well, about feat m I'm like02:11:29
just said02:11:35
no I won't draw graphics hd vsky02:11:39
why why should I do this02:11:42
we calculated this value and realized that02:11:45
looks lame they took the function02:11:48
which will represent this quantity02:11:52
that's okay let's wear a mask02:11:56
from each number, that is, we write down what02:11:58
module m is the same as what02:12:02
degree, so we check that you didn’t screw up, but02:12:05
too many people02:12:08
to what extent will it work here?02:12:15
sir tempo is defined only in integers02:12:19
in numbers how can it even be02:12:20
continuous what are you talking about02:12:22
even as natural as she can be02:12:26
smooth continuous if defined02:12:28
only in natural numbers02:12:31
change the schedule not ddddd pt all02:12:35
good in general people are alive Rajiv 5 m02:12:40
of course, the fact is that there are 5 numbers02:12:43
kapoor02:12:45
arctic not 502:12:47
multiply by x1 and x2 and so on x02:12:51
fiat index02:12:55
super alive02:12:58
whoever is alive I like and02:13:02
So02:13:04
just like that, look how beautiful it is02:13:07
common factor let's write the difference02:13:10
since these two expressions are comparable02:13:13
this means the difference is divided by m02:13:16
now I will write more about02:13:31
dealer prices and02:13:36
really x and you can put it in foil and eat it02:13:38
absolutely true denis of course02:13:43
x1 x2 and so on x index we take out02:13:48
out of brackets and02:13:49
we get a to the power02:13:57
-1 are divided by m02:14:06
the most important moment guys is now me02:14:10
I can conclude that this is the thing02:14:12
reciprocally just with02:14:14
because in the given system of deductions I02:14:17
I specifically left only those numbers02:14:19
which are relatively prime s.m.02:14:22
means their products of common factors02:14:25
Sam doesn't have02:14:29
thing and m are mutually prime means here02:14:33
find all the factors of the number m and02:14:37
So a to the power of 5 minus 1 is divided by02:14:42
And02:14:44
this is called a theorem02:14:56
n102:14:58
Well, of course, who will stop you from recording02:15:00
just more familiar from a larger number02:15:03
subtract less02:15:07
what is more interesting for you is the sharpened combo02:15:12
I don’t choose short between mom and dad02:15:16
answer02:15:20
yes yes yes how about let me copy my homework02:15:24
yes, just not exactly02:15:25
proof02:15:27
we are absolutely the same, just everywhere02:15:30
we need to replace p minus 1 with 5 in general02:15:34
nothing has changed, just need to02:15:37
the prize had to be taken not specifically the number02:15:39
12 and so on n minus 1 and some02:15:41
numbers x 1 x 2 x with index feat m then02:15:44
there is some kind of reduced system02:15:47
deductions02:15:50
that's02:15:56
that's not hot enough for this room02:16:00
so well, we have proven Miller's dash02:16:04
emotions02:16:06
hurray let's rejoice02:16:12
questions if there are any02:16:16
let's wait a little02:16:22
small company topic is a consequence of the topic02:16:24
Euler for m equal pc for prime02:16:27
numbers n02:16:34
well fear dm for the number 10 it’s not nine02:16:40
it’s true, that’s what we did here, well, fedot02:16:43
I think they missed this one after02:16:45
they came a moment later and I said that02:16:48
paraphrase five minutes late themselves02:16:51
let's just say but02:16:52
you can just watch this part here02:16:55
even she is almost entirely left here we are02:16:57
counted how many numbers we don't have02:16:59
superior in number reciprocally simple with him02:17:01
A02:17:04
no it won't be great02:17:12
V02:17:14
prove to them02:17:16
well look guys let's do it now02:17:20
we will prove one more theorem and then02:17:24
Now let's solve the problems as I said02:17:27
already but most likely either half past one or02:17:29
at two o'clock we will start the second block bye then02:17:31
let's make it a little smaller and basically02:17:34
It's OK02:17:35
there won't be anything wrong with that02:17:38
now we will prove02:17:40
Also the Wilson theme here is so cool02:17:44
Wilson's theorem is not like that02:17:46
popular as a small theorem of course02:17:48
farm02:17:49
Here02:17:53
Well, I'm erasing everything, I hope that someone needs it02:17:57
those the screen did but like me already02:18:00
said the principle of the abstract km and you02:18:02
I won’t post it right away, but in about five days02:18:05
he will02:18:06
yes yes yes we are talking about reverse bets now02:18:09
let's talk too02:18:14
cream and mills02:18:22
A02:18:23
Maxim also say that everyone is watching yes yes02:18:27
yes, you got caught, maxim that you are all02:18:29
listen yes you need to02:18:31
all this was to check02:18:34
Maxima Rot Gold listens to me too02:18:36
it turned out that no02:18:43
I already said yes what02:18:46
there will be a summary02:18:49
so yeah02:18:51
wait here or mobilz, remember we by and02:18:54
-1 factorial was taken out of brackets and02:18:58
this kind of thing doesn’t matter what it appeared02:19:01
one factorial it reduces heat not02:19:03
contains no common factors p02:19:07
so the funny thing is that02:19:11
actually we know02:19:14
by and -1 factorial02:19:17
comparable to minus one modulo02:19:29
for and02:19:31
there is a mustache for biology already for chemistry and02:19:37
g was definitely 12 o'clock with straight and02:19:41
probably if you look in the main02:19:43
school group subscribe and it will come to you02:19:47
news when will it be02:19:50
corresponding to its own02:19:52
chemistry02:20:03
which02:20:04
understand nothing02:20:06
not one from the very beginning, but that is, I’m there02:20:09
explained what modulo comparison is02:20:11
It’s clear enough I repeat it a bunch of times02:20:15
explains02:20:16
Here02:20:20
Means02:20:22
Gregory simply formulated simply02:20:25
formulated a02:20:31
you can't make mistakes in Russian sage02:20:34
ruins everything without a soft sign02:20:38
yes, it must be a prime number02:20:41
in general, almost always, yes, I’m super, I’m02:20:45
have a prime number02:20:46
here you need to clarify02:20:51
So02:20:52
[applause]02:20:57
let's prove it02:20:59
pretty cool proof I will02:21:02
use the following interesting fact02:21:04
look we took prs in02:21:08
chose PSV02:21:14
A02:21:16
1 2 3 and so on through and -102:21:21
multiplied by a02:21:24
received02:21:26
a2a for a and so on through and minus 102:21:31
multiply by02:21:34
so this one is small02:21:38
we need a new hero02:21:52
but is it really like that in the head?02:21:56
it's true but you have to understand02:22:03
a big request to the guys who are now02:22:05
looks further will look try02:22:07
strongly understand they cram the moves that02:22:11
we do understand why we are here at all02:22:13
we do such things, it’s quite possible02:22:16
do02:22:18
Here02:22:20
[music]02:22:22
look let's02:22:25
let's think about where the one is here02:22:50
A02:22:51
well, I can’t specifically answer where she is here02:22:54
I can and you know for sure that she is here02:22:59
I know for sure that one of these numbers02:23:02
gives remainder 102:23:05
it's true02:23:09
well because we took the one given02:23:12
the deduction system was multiplied by and we02:23:14
we understand that again the reduced system02:23:16
deductions means somewhere a witness of these02:23:19
numbers must have remainder 102:23:22
But02:23:24
I don’t know where but it is there, that is for02:23:28
some b.a.02:23:36
[applause]02:23:43
and Bine02:23:46
ayran b is comparable to one in modulus02:23:49
P02:23:57
don't let me write fractions now02:24:00
yes I don’t know what 1 is from and helped p02:24:03
in general, integers are strong after all02:24:06
that is, we have now proven02:24:11
proved that for an arbitrary number a02:24:13
which is not divisible by p is like this02:24:16
called reciprocal remainder02:24:19
b02:24:21
remainder02:24:23
reverse as02:24:29
that is, such that their product is equal02:24:32
one02:24:34
By the way, this is the only remnant02:24:37
that is, b are the only ones why but02:24:42
when I multiplied by a only one of02:24:43
numbers turned to one02:24:47
two numbers cannot turn into one02:24:50
could have done what I didn't succeed02:24:52
reduced system of deductions and I already02:24:53
PSV proved it02:24:56
By the way02:24:59
he's the only one02:25:17
let's give it to moms, it's really possible to take it02:25:20
specifically b is equal to a to the power of b minus 202:25:23
this tells us little about the company, but02:25:27
it’s as if you can understand it even before it02:25:36
that is, there are inverses to the number02:25:40
extracted from numbers 12 and so on to -102:25:43
beta some remainder02:25:48
so simple and really look02:25:51
it turns out if this is the reverse remainder a02:25:55
then and this is the remainder of the inverse kb because02:25:58
and b&b is the same thing02:26:02
that is, if a02:26:04
back if this is the remainder inverse a02:26:07
then vice versa and this is the remainder of the inverse kb02:26:12
that's why they say that these are mutually inverse02:26:14
leftovers02:26:18
In fact02:26:23
maybe it will happen then the rest will be02:26:28
this is itself02:26:31
this could happen02:26:33
let's answer this question here02:26:36
can02:26:40
she02:26:42
be comparable to one modulo n02:26:47
generally yes but not very often02:26:55
and mtf direction this is in any direction02:26:57
side you are going to and remove no02:27:00
you can't, I forbid you to extract the root02:27:04
no two ones minus ones until so02:27:09
it happens but I would like to understand02:27:11
find all the numbers for which I have02:27:14
there is well for which the inverse02:27:17
the remainder is the number itself is perfect02:27:19
it is obvious that a is equal to one or minus02:27:22
1 fits, why not others?02:27:24
leftovers, by the way, let me get you02:27:26
I'll ask why 1 and minus 1?02:27:31
the only exception02:27:32
that is, when02:27:35
we'll give the rest to ourselves02:27:52
just a minute02:28:02
so what 191 blows on x plus or minus 1 o02:28:06
why did you decide that there is only02:28:07
Plus or minus 1 should be a little02:28:10
you never know02:28:14
Well, I'll explain why I do this02:28:18
I stand for accurate evidence here02:28:22
For example02:28:23
5 squared02:28:24
five by five02:28:27
should be comparable to one modulo 802:28:35
although a five is not plus minus one02:28:38
Of course you will say that this is an eight and02:28:40
not a prime number but we are now02:28:42
consider a prime number but it's like02:28:45
minimum means to prove something02:28:47
it is necessary that is, yes, these are not prime numbers02:28:50
in general it's just not true02:28:54
by the way it means something is wrong02:28:57
obviously that means yes02:29:00
Evgeny writes correctly02:29:04
units write correctly and well done to Igor02:29:07
in general, who didn’t see all those well done02:29:10
correctly suggest to do the following02:29:12
write difference of squares hello02:29:14
Hello02:29:16
let's just write the square minus 1 divided02:29:19
on p true02:29:21
it says here and a square is comparable to02:29:23
unit modulo ps means a square02:29:25
minus 1 that is, different is divided by p then02:29:28
such a difference at the withers is a minus 1 per02:29:32
plus 102:29:35
this product is divisible by p02:29:40
means and because a02:29:44
minus 1 is divided by p02:29:48
or a plus 1 is divided by p02:29:53
Well, in this case, the protection of the mask is limited to02:29:57
modulo n in this case a is comparable to02:30:00
minus one modulo all02:30:09
painted the difference received something before or02:30:13
1 is equal to or 2 is divisible by p means02:30:17
either this or this way you realize that on02:30:20
actually02:30:21
that's why I wrote it for this02:30:24
Prime number02:30:28
well because for not a prime number y02:30:31
me not necessarily one of the brackets02:30:33
shares it entirely, but if I am for 502:30:37
squared minus once I would write02:30:38
it turned out 5 minus 1 by 5 plus 1 4 by 602:30:42
neither one nor the other is divisible by eight02:30:44
the product of eight is divisible by02:30:47
that would be the trick02:30:50
no, of course02:30:52
we can assume that these numbers are02:30:57
part a factorial but what if I take02:30:59
and less than p then why not now02:31:02
this is roughly what we will do02:31:04
let us state once again that02:31:07
we have now proven we have proven02:31:10
what for each number02:31:13
there is a reciprocal remainder for02:31:16
every number that is not divisible by p02:31:18
of course02:31:19
we proved that he is the only one02:31:23
proved that if the inverse of a then a02:31:27
inverse kb and we have proven that the number02:31:31
maybe back to yourself only02:31:34
when it's one or minus one02:31:43
step02:31:45
determined how and I'm trying to understand the type02:31:49
it could either be like hell if not02:31:52
redefined a02:31:56
I'm just looking into this case02:32:01
ok so now it's important we're almost02:32:05
actually proved the uniqueness of I02:32:07
has already proven02:32:08
I've already proven it and it's good for her to skip02:32:12
remember I said about the fact that we multiply02:32:16
the reduced system of deductions for a and yes and02:32:20
then I can't be among these02:32:24
numbers 2 ones02:32:29
we have already proven this with you, but what02:32:32
it turned out to be a reduced system, what are you02:32:33
that means 2 and there can be no game here02:32:38
the inverse remainder is the remainder02:32:41
the remainder b2b is comparable to one in absolute value02:32:44
P02:32:45
well yes for equal to 1 minus 1 it is obvious that02:32:49
this thing will be comparable to unit 102:32:52
squared is one minus 1 squared02:32:54
this is also 1 but for the rest it is not02:32:57
works in a simple way from Luther and so02:33:01
this means but let's look at02:33:02
module 7 for example02:33:05
how do remainders behave modulo this02:33:09
since we write all these pairs mutually02:33:12
back to02:33:14
sta-2 and 402:33:20
helped02:33:24
73 how much should you multiply by 502:33:29
get unit modulo02:33:35
7102:33:37
sent pair with unit02:33:42
-1 is sent in pair with minus one02:33:54
that is, look, all the orcs are divided into02:33:58
couples mutually back because02:34:02
there is a unique and inverse02:34:06
side works too02:34:12
therefore for each number there is a well02:34:16
that is why everything is divided into pairs02:34:18
true one corresponds to itself02:34:21
minus one also corresponds to itself02:34:23
to myself02:34:25
let's take a closer look then02:34:28
product p minus 1 factorial what is it02:34:30
such02:34:37
product one by two by three and so on02:34:40
Further02:34:42
I can break everything into pairs mutually02:34:46
reverse balances02:34:54
Airborne Division02:34:57
somewhere02:35:00
couples02:35:01
mutually inverse astatine and02:35:11
The main thing is that there is no steam left02:35:15
this and02:35:20
chika and minus one02:35:33
Well, look, in every pair we have02:35:37
product gives remainder 102:35:41
each of these numbers02:35:45
comparable to unity modulo n02:35:49
also comparable to unity mod n02:35:52
until always remains because I already02:35:56
explained02:35:57
if it's p minus 1 I just don't want to02:35:59
write n well why did I already prove what02:36:04
minus one is always back to itself02:36:07
well, minus 1 squared is one02:36:10
therefore they always remain one minus02:36:13
one02:36:17
everything else is steamed02:36:20
Well, what can we compare it to then?02:36:28
why -1 is always the remainder well minus one02:36:33
in the given system there are yes02:36:35
it's simple but p minus 1 you perceive02:36:37
if ok if you don't like minus02:36:40
units speak by and minus one02:36:44
this is comparable to minus one02:36:55
why did the rest get together in pairs?02:36:58
one more time for each number there is02:37:01
reciprocal remainder we proved it02:37:05
for the remainder and I can find mutually02:37:08
reverse remainder b among the rest and02:37:11
and you don’t need to look for anything02:37:13
its reciprocals the remainder is a02:37:17
I took 16 hours 2 found for him02:37:21
back02:37:24
I saw the remnants and they fit02:37:27
same with everyone02:37:32
in the end it turns out that it is comparable02:37:35
with minus one02:37:36
modulo02:37:43
Thus I proved by n minus 1 real02:37:49
comparable to minus one but either02:37:52
let's write it differently plus 1 is divided by p02:37:57
This is called Wilson's theorem02:38:01
Wilson's theorem02:38:10
we may not be used as often as02:38:12
the little farm theorem on its own02:38:13
very beautiful and statement02:38:20
well, in practice, I already said yes02:38:23
there are not many tasks with it02:38:25
are resolved, but sometimes they meet, and what does it matter?02:38:28
Well, if I don’t know, what the hell can you prove?02:38:31
but I should rather pay your attention02:38:34
guys' attention to02:38:36
[music]02:38:39
it is the idea of reverse remainders that can be02:38:43
It’s even more useful to bring up something other than the topic itself02:38:45
Wilson yes, but what for each number02:38:48
there is an inverse remainder for each02:38:51
the remainder exists back02:38:53
there really is a situation when02:38:56
let's turn this remainder back to ourselves02:38:59
one minus tit there are no others like that02:39:01
It happens02:39:04
can be found I already told Dima that yes by02:39:08
such a formula is possible if p is composite02:39:10
can find using the formula from degree and from m02:39:14
minus 102:39:16
by or multiply the reduced system02:39:19
deductions that is, look for reverse rates02:39:22
not really that convenient02:39:24
the pleasure is so-so02:39:27
but in principle, in principle, we have proven that02:39:30
they are02:39:33
we take the rest and02:39:36
from02:39:40
Well02:39:42
sense that your task is like this, that is, I02:39:46
in general specifically by and minus 1 factorial02:39:48
consider now02:39:50
specifically now I was considering exactly02:39:52
at -1 the factorial is as if the same02:39:55
Just02:39:58
maybe than uh huh well you know what02:40:01
when you are looking for a solution to ice there is also like02:40:05
wouldn't run smoothly in some sense02:40:09
Euclidean algorithm is great but I don't02:40:13
said that he does not frida count on02:40:15
every step what you have left up to02:40:18
Honestly, writing a formula for this is easier than02:40:20
overkill couldn't be simpler02:40:27
we all proved wilson's theme let's02:40:31
Now let's solve a few problems for everything02:40:34
this business02:40:35
Here02:40:38
Means02:40:41
Let's solve problems using Euler's theorem02:40:48
path02:40:52
I'll only find an hour here with me02:40:55
in leaflets I love my printer which02:40:58
he just takes a bunch of leaves and eats it for him02:41:03
neglect it altogether02:41:06
so with that ready let's move on02:41:09
the third page he02:41:12
Euler theme02:41:16
so if sk problem on this topic02:41:19
we will be glad that you have a mailer02:41:22
really helps02:41:27
I didn’t find any questions in the chat because02:41:30
that guys don't02:41:45
[ __ ]02:41:47
no, maybe it doesn’t feed the task, I know02:41:51
they are in front of me, secondly I won’t02:41:54
all02:41:55
embroider02:41:59
here it is02:42:01
that means, well, yes, but it’s not on this piece of paper02:42:04
what you can find on your own02:42:10
the definition of the lehr function is there02:42:12
Euler's theorem formulated yes yes yes02:42:15
yes yes yes this is this one02:42:18
if you feel it then write02:42:20
What exactly do you want02:42:23
so let's take a little look now02:42:25
how does Euler's theorem work02:42:27
there are formulas for and from m02:42:31
in principle yes it kind of exists it02:42:35
We don’t have any formulas to prove right now02:42:37
let's do it for us now the candy is not very good02:42:40
Interesting02:42:42
[music]02:42:43
but let's solve the second problem02:42:47
I think the tasks are very cool and kind of02:42:50
not a word about Elera wrong chat and02:42:54
a normal task which nevertheless02:42:56
can be solved perfectly with the help02:42:58
Euler's theorems02:43:01
don't whine about this yet02:43:05
there are fewer people, that's the first thing02:43:06
normal as if it’s hard for someone02:43:10
withstand more there 200 hours for02:43:14
it’s hard for someone, in principle, yes, as it were02:43:17
some topics to be aware of, well, from the type and this02:43:21
the Olympics are here02:43:24
no need to chat about this simpleton02:43:27
write02:43:30
So02:43:31
the most important thing is that those who remain02:43:34
understood everything and02:43:37
I will try to answer all your questions02:43:40
answer quickly so02:43:44
It's OK02:43:46
let there be a few people left there but02:43:49
which will actually work like this02:43:52
prove that by number 2 in 201802:43:58
3 3 pages of document from the first link02:44:02
under the description in the description under the video02:44:05
task number 202:44:08
prove that the number 2 in 2018 is possible02:44:12
add a few numbers to the left like this02:44:15
to get a power of two again02:44:20
on the left, well, you know, distant 2 2005 there02:44:25
there are a lot of numbers written in this case02:44:29
there are a lot of numbers, this is the number that02:44:32
equals 20 thousand 18 and we are told that02:44:36
on the left you can assign several numbers02:44:38
to get a power of two again02:44:47
Here02:44:56
let's formulate02:44:58
what is he asking us to prove?02:45:03
like damn02:45:05
this way it looks like cinnamon02:45:08
all the magic really is this02:45:10
asks us to prove if we talk about02:45:13
the rest, let's transfer everything to02:45:15
remnant language02:45:17
then it’s actually easier for us02:45:19
prove02:45:24
let Dima say correctly that it’s like02:45:26
It’s not clear how this is related at all to02:45:29
some creams and lera leftovers02:45:31
comparison so on02:45:35
who understands how this is still possible02:45:40
connect with02:45:41
leftovers02:45:47
Arseny but if he didn’t do that probably02:45:50
it means that I just don’t really want to because02:45:52
that I don’t want to carry around all sorts of numbers02:45:54
one by two and so on and how are you a a n d02:45:58
but why02:46:05
composition com on 10 is somehow related02:46:13
the sum of the numbers has nothing to do with it at all if02:46:15
Honestly02:46:19
take the rest02:46:23
let's do it this way02:46:27
let tour cards02:46:30
no let me here02:46:39
what do I want to be done for 2 in02:46:43
the card must be executed for 2 in02:46:46
ride02:46:47
so that I can attribute to the number 2 in 201802:46:52
a few numbers to the left and get two in02:46:55
roll it, what does the sum of numbers have to do with me02:46:58
I really don't understand02:47:00
worry about the sum of the numbers off02:47:10
vase02:47:12
I'm filming02:47:14
We don’t have a binary number system here02:47:17
in no case will it help because well02:47:18
numbers must be written in decimal02:47:21
the number system is true02:47:24
this is where it actually begins02:47:28
begins to appear to be absolutely poor02:47:31
comments let's understand that we02:47:35
they just ask you to find something like that 2 in ka02:47:39
give02:47:40
comparable to 2 in 201802:47:44
modulo n will help 10 to the power m02:47:53
I need to find something like this02:47:58
let's understand why this is enough02:48:01
do02:48:03
look, let this be done02:48:08
modulus 10 to the power m is exactly02:48:11
last n digits02:48:13
the number gives the same remainder when divided02:48:15
by 10 to the power n which is the number02:48:19
formed by the last digits02:48:31
module 10 power n well because02:48:33
the sign of divisibility is that the last n02:48:36
the numbers are actually a 10V module02:48:38
degree n02:48:40
Here02:48:42
Well, if we find something like that, then naturally02:48:46
if it already ends with the right ones02:48:49
numbers are not a problem, just assign them02:48:52
the remaining yes02:48:54
that is, now we have a task02:48:56
formulated normally normally according to02:49:00
humanly sorry02:49:08
that you haven't heard02:49:10
try to prove02:49:12
in principle but02:49:17
why is this true if we subtract the number02:49:20
formed by the last digits02:49:22
the result is a number x that ends02:49:24
by n zeros it is divisible by 10 to the power of n02:49:29
well that's why02:49:31
the remainders are the same for number and number02:49:34
formed by the last digit02:49:38
no but let's cut it short for now02:49:41
let's not be more careful more careful02:49:44
abbreviations meaning the way you speak02:49:47
yes, but I won’t do that so that no one02:49:49
didn't understand me02:49:50
it's wrong to shorten so I better not02:49:53
will02:49:54
come on, let's really do better02:49:57
let's write the difference as something new02:50:00
calmer more familiar02:50:02
I want this difference02:50:06
marveled at 10 to the power n02:50:15
well, good, can you have two more food 1802:50:19
endure02:50:35
A02:50:39
10 to the power m is actually the same02:50:43
2 to the power n by 5 to the power r02:50:48
we can separately02:50:52
prove divisibility by 2 in Beijing02:50:55
separately by 5th degree n02:50:58
Well, about 2 to the power of n, who understands02:51:03
how will it work here why02:51:06
the left side is divisible by 2 to the power of n02:51:17
ask two years obviously task card go02:51:21
years but he 2 roll02:51:24
mtf has nothing to do with it at all02:51:30
so it’s clear what this is for02:51:33
there will be no sharing at all02:51:35
that's right02:51:37
this will be on the 2nd degree because n02:51:42
less than 2018 really look and02:51:45
for this number of digits in the number 2 in 201802:51:49
obviously 2 in 201802:51:52
less than ten 201802:52:01
but from02:52:03
17 is the number that contains 201902:52:06
types the first number which contains 201902:52:09
therefore, number 2 in 2018 is less than 201902:52:14
no more than 201802:52:18
it is obvious that science02:52:22
number 2 in 2018 less than 201802:52:28
cypher this is what is written here02:52:32
it turns out this thing is divided into 2 in02:52:34
degree n02:52:36
got this sorted out02:52:40
something needs to be divisible by 5V02:52:43
degree n is unlikely to help us 2 in 201802:52:47
that means I want02:52:50
I want to02:52:53
she02:52:56
shared02:52:59
on02:53:00
5 to the power n02:53:08
that is, it’s not bad that 2 powers k minus02:53:14
201802:53:16
-1 is divisible by 5 to the power n if I02:53:20
I will prove that such a thing exists, that everything02:53:22
problem solved02:53:29
if I prove this to the point it is02:53:32
problem solved02:53:34
well let's rewrite this as a simile02:53:39
I want this thing to be comparable02:53:42
with unit modulo 5 to the power n and02:53:47
why does this happen02:53:49
why does this exist02:54:02
2 thousand02:54:04
two and four are not comparable to one02:54:06
modulo 5 to the power n, more precisely modulo02:54:08
five and lipstick 5 to the power of n we want already02:54:10
we want divisibility by powers of 5 n to get02:54:13
for the mailer's loss, the mailer's loss of course02:54:18
on the topic of peace02:54:28
2 degrees and02:54:31
o 5 to the power n is comparable to one in02:54:35
module five degree here I am of course02:54:37
I use the fact that two and 5 are to the power of n02:54:41
these are coprime numbers guys02:54:43
don't forget to check the company02:54:45
indivisibility in the topic pocket about prison or02:54:48
you need to check mutual primality02:54:51
without this it would be impossible02:54:54
so you can just take it02:54:57
for some reason they are 4 guys02:55:00
denis02:55:02
if this is a four then it will be 2 of 402:55:05
it is not comparable to one modulo five02:55:07
degree m it is comparable modulo five but02:55:10
because we need 5 degrees n true02:55:16
so we can take it02:55:19
well, for example, equal to 5 to the power m02:55:25
plus 200802:55:31
for example like this02:55:37
everyone has proven that there is, well, for example02:55:39
you can take this one02:55:41
then the difference will be equal to 55 powers of n02:55:48
well then that is what you have to do with it02:55:52
mailer if I take this to then I have achieved02:55:55
the fact that the difference is equal to pi from 5 in02:55:58
degree n, this degree is equal to and from 5 to02:56:02
degree n simply by Euler losses02:56:07
any number from the Ph.D. degree comparable to02:56:12
and de pert of degree 5 n is comparable to02:56:15
unit modulo five power n if a02:56:18
and 5th power n are mutually simple to apply02:56:21
for Aries in deuces02:56:24
2 to the power of fett 5 to the power of n is comparable to02:56:28
units for water 5th degree02:56:31
Miller's mystery can be proven for real02:56:34
in fact, in principle it is possible, that is, then it is necessary02:56:37
prove that there are still leftovers02:56:40
loop modulo 5 to the power n02:56:43
something really doesn't stick to02:56:44
unit but it’s kind of so sickly02:56:48
theory is still needed for this02:56:58
so let's climb in, let's wonder who02:57:03
understood who understood let's put a key to02:57:07
dinner, how to learn it, yes cha-da02:57:13
there are no problems with the file, what will I do?02:57:17
it will be possible to disassemble the notes, but it’s straight02:57:20
everything, everything is unlikely, but damn it’s like there02:57:23
more than a hundred olympiad problems02:57:26
so the great Fedot didn’t understand02:57:30
about what is unclear, you ask02:57:34
specifically where the task will come from02:57:38
some listen you don’t need to perceive02:57:40
what is the task there anyway?02:57:42
some02:57:44
Lomonosov or there are no problems yet, I seem to02:57:49
doesn't know the exact source02:57:53
so but you climbed the massive keys02:57:56
put it cool if any02:57:58
If you have specific questions, please ask02:58:00
simple but02:58:02
for entrance to Chinese kindergarten02:58:05
so now I’m still looking at which one of these02:58:08
problems, you can take problem 4 very02:58:11
good comprehension task02:58:17
what's wrong with02:58:21
problems from the Olympiad 239 it started02:58:26
What's the height for?02:58:29
Well02:58:39
I don't know02:58:42
like 902:58:54
hello again squid02:58:57
wrote turgor02:59:01
12 or a yes yes yes yes yes yes yes02:59:09
So02:59:12
but let's understand02:59:15
there will be an idea for the next leaflet for 2202:59:18
here in this in this there is not even a leaf on02:59:20
my you just did homework simple02:59:23
number 239 to me02:59:26
here is the 4th task for how many does it mean02:59:29
the meaning of the number and where is one02:59:34
well where not from 1 to 100002:59:40
there is a number02:59:42
then 2 to the power02:59:44
minus 102:59:47
is divided by it02:59:51
for how many and02:59:56
there is g from 1 to 1000 which is 2 in03:00:01
power minus 1 is divided by and03:00:07
good luck to you Dima03:00:11
show or have nothing to do with it03:00:19
the indicators have nothing to do with it yet03:00:23
so first of all let's start with this03:00:26
there is no question for which ones03:00:30
for which and 2 to the power minus 1 is definitely not03:00:34
can be divided into and03:00:45
Denis immediately speaks Freddy Ng03:00:50
rap for blacks for what it's obvious yes yes03:00:56
even numbers03:00:58
immediately there is a contradiction and03:01:02
definitely odd03:01:06
that for a black number we are from black03:01:10
subtract one and get odd03:01:11
bears on cannot be divided by even03:01:15
live well03:01:18
some number but open the conditions03:01:22
file with conditions03:01:23
for how many values and from 1 to 100003:01:28
There are numbers from 1 to 1000 tokens03:01:34
a number from 1 to 1000 some question03:01:37
for how many does such a thing exist?03:01:40
exactly odd03:01:46
that a for odd numbers exists03:01:49
this doesn't exist03:01:53
We killed part of the problem and realized that the even numbers03:01:57
not suitable for odd and exists03:02:01
the same03:02:10
I'm not sure it's always 10 Igor03:02:27
there is a skin hopper03:02:28
thank you where to speak we post and you don’t tell me03:02:31
enchanted03:02:49
Igor 1 hour or so i me03:02:56
Alexander Delo says thank you bye03:03:00
there are jokes there joke Alexander says business03:03:02
look if it's an odd number03:03:05
let's reformulate this condition as03:03:08
comparison of what I want I want 2 in03:03:11
same degree03:03:12
was comparable to unity in modulus and03:03:19
but it definitely fits03:03:28
whether03:03:31
equal and here's a mini03:03:35
let the computer want it03:03:40
let's not disappoint him03:03:47
but it actually turns out exactly right03:03:49
Euler's theorem turns out 2 not I'm not03:03:54
comparable to unity in modulus and03:03:57
if two and and are coprime no but03:04:02
I could have missed all this03:04:05
I see he’s winning, I’m talking about what’s started03:04:10
Here03:04:11
that is, they just took it and lived equal to 5 and03:04:15
can I take it? I have conditions03:04:18
that fat should also be about 1 to 100003:04:21
Truth03:04:23
there is such a condition, well, also 1 to 100003:04:36
will fit below this condition I03:04:40
I take it I can't it won't be from 1 to03:04:43
100003:04:48
but not enough, they really are not03:04:52
this03:04:54
tells us the matter03:04:57
so what?03:04:59
let's realize that fiat and nothing more03:05:02
than usually even strictly less if03:05:06
just not equal to one03:05:12
well, it’s clear what offers are03:05:15
number of natural numbers from 1 year and03:05:17
mutually prime and well, that is, we are from and03:05:21
Some numbers have been removed, some remain03:05:24
obviously no more than03:05:26
it doesn't exceed a thousand and that's obvious03:05:30
that there is at least one such thing, for example03:05:32
units03:05:33
that's why everything works out03:05:38
for every odd number and so on03:05:41
there will be03:05:44
hurray the problem is now completely solved03:05:48
that is, the answer is for 500 numbers or 500 as03:05:53
Right03:05:56
for 500 numbers there is such a thing, that's all03:06:00
odd numbers numbers03:06:03
uranium problem solved03:06:08
around these if there are03:06:16
500 what03:06:18
Fine03:06:21
so write if we have any questions03:06:26
Now let's look at the problem03:06:27
finally a cool eighth for example03:06:33
eighth03:06:38
or take the fifth one apart03:06:41
maybe pick up 5 yesterday03:06:45
5 cool03:06:48
1 size03:06:56
exactly less where less meaning less03:07:00
anyone but everywhere I write in the wrong way03:07:04
5 cool do in the description kuzma if u03:07:10
you are not the first link in the description03:07:11
update as much as possible givati added there03:07:14
materials03:07:19
Problem 5 yes I agree now let’s solve the fifth03:07:23
need 1310 I kind of said the date that we03:07:26
We'll take a little longer to solve this block03:07:28
In principle, it’s even good that we still have strength03:07:30
it's okay if we're almost there03:07:33
four hours straight of work, just a little ready03:07:36
the duration will be a little shorter03:07:39
further we still have to wait wow, what damn topics I03:07:42
I don’t know how I can do everything in six hours03:07:45
tell03:07:46
after when03:07:52
and three o'clock03:07:55
celebrating myself03:08:04
so 5 task archer axial condition on03:08:08
actually03:08:09
prove that for any natural03:08:11
number n, the number can be found by sum of digits03:08:15
equal to n03:08:24
which is divisible by m03:08:29
yes yes yes primary that's when we finish03:08:32
Let's do the first block for now I think03:08:35
it’s still normal, especially since we’ve just gotten there03:08:38
to the berries, yes, we are now starting to use03:08:41
Euler's theorem which came before03:08:43
proves it might seem like a03:08:46
[ __ ] it, we need everything, but here’s this for you03:08:49
it seems like a very useful thing03:08:52
allows you to solve really beautiful problems03:08:56
here, but let's think a little, I need03:09:00
take a sip of moto03:09:40
yes iron pipe iron pipe in this03:09:42
such03:09:45
very tasty very tasty03:09:50
you can google print like this03:09:58
I was given this Thomas seven years ago03:10:03
By the way, it’s exactly 7, well, almost for my birthday03:10:09
children whom I taught who are now on03:10:12
fifth year03:10:14
which one should I leave?03:10:20
waters of malta romantic yes03:10:25
there is a suggestion that any numbers can be used03:10:28
write the sum on and then on the right 0 and03:10:31
add it well03:10:32
as if you write 0 on the right and03:10:35
you can develop that divisibility by03:10:37
power of two power of five get03:10:41
if relatively speaking03:10:43
the number is not divisible by 7 and will not be03:10:48
divisible by 7 if you write any03:10:50
numbers like this are only powers of two03:10:52
powers of five03:10:55
you can get03:10:58
no but it's like moisture it's me on it03:11:02
I brew for 12 hours every class03:11:05
after all, it doesn’t require some03:11:09
You won't be able to drink enough energy stuff03:11:12
eventually poison the body03:11:18
so why let's get another quinta of ideas03:11:21
in general, in fact, just add it03:11:23
zeros on the right are good this is real03:11:26
ok well I mean at least like03:11:29
yes, if n03:11:32
we will expand as 2 to the power of alpha by 5 to03:11:37
powers b multiplied by k03:11:40
then like divisibility by this thing I03:11:43
I’ll just get there by adding the end of the bunch03:11:46
zeros03:11:47
the real problem is getting03:11:50
divisibility by k03:11:57
here it is03:11:59
number of digits03:12:02
good idea, hello Anastasia03:12:05
let's understand our problem03:12:10
actually03:12:14
let's get the number c for now03:12:17
the sum of digits and equal to which are divided03:12:20
on to03:12:26
Let's fill the top five for now, let's decide03:12:29
problem with numbers03:12:31
let's start with him03:12:35
Well, just name the number right away03:12:38
divisible by k sum of digits k their knows them03:12:44
knows really doesn't know about about03:12:48
Artem vitalich or Alexander Romanovich03:12:51
but according to Newton's second law03:12:54
hello Alexander Romanovich of course03:12:57
yes because but only such Heine03:12:59
how can Alexander Ivanovich pretend03:13:00
Artyom Vitaly how did I figure you out? I got you03:13:03
figured out checkmate03:13:06
Hello Alexander Ivanovich03:13:17
A03:13:19
And03:13:21
here not here just with us and03:13:25
so yeah03:13:29
Well, let's solve this simplest problem03:13:34
focused on the task well cool thoughts03:13:37
interesting and beautiful and well at least03:13:40
conditions are exciting03:13:42
engagement in one and a half lines is very03:13:45
understandable in simple language03:13:47
the solution will be using Euler's theorem03:13:52
Well03:13:54
nine we're lucky, you know, it's like03:13:57
let's take to but 17903:14:01
knows how to look for them03:14:05
that is, no, but what should you take?03:14:09
which is divisible by nine we need for03:14:10
prove this for each number n, you understand03:14:13
for each number n03:14:16
so we can't choose the number ourselves03:14:19
n if we were choosing I would choose m equals 103:14:22
and would say goodbye to everyone03:14:25
tasks are obvious but they gave us03:14:29
To do this, they must prove that03:14:30
there is a number with a sum of digits that03:14:33
divided by03:14:35
not 170 a lot of zeros won't work because03:14:38
that the sum of the digits of this number is not 17903:14:42
no problem getting divisible by 17903:14:47
no problem getting the sum of the numbers 17903:14:51
problems getting both03:14:53
simultaneously03:14:55
Here03:14:59
In general, I suggest you do it, look03:15:03
let me come up with many many numbers with03:15:08
the sum of the digits 1 which gives a remainder of 1 when03:15:13
dividing by03:15:17
k.k. what will I leave in the number n if it is two and03:15:20
give out an A03:15:24
it means I want about my desires I need03:15:28
I want to pick up loudly03:15:31
a lot of numbers03:15:36
with the sum of the digits 1 and03:15:44
A03:15:46
starcom03:15:49
anine modulo k03:15:53
I want a car wash03:16:00
Igor, we're probably lucky that03:16:04
2007 is divided by the sum of the numbers understand03:16:09
if the number itself is not divisible by its sum03:16:12
digits for the number 179 or 239 I assure you03:16:15
it's not like that it doesn't share03:16:19
there seem to be some problems03:16:24
you won't do that03:16:27
so you understand why I want this03:16:31
let's not mention the numbers yet03:16:35
let's understand why it will be cool03:16:39
probably because we can then03:16:42
fold if we're lucky and there03:16:45
the sum of the numbers will also simply add up by taking03:16:49
such k numbers03:16:51
we will get what we want03:16:56
look for now I just stated about03:17:01
I want to write a lot03:17:04
numbers with a sum of digits of 1 I think you03:17:08
guess what these numbers are as soon as possible03:17:11
In general, numbers have cp1 itself and a remainder of 103:17:14
modulo to if you take add a lot03:17:17
there are k such numbers then look at the sum03:17:22
I hope the numbers add up and what doesn’t03:17:25
will be a transition through the discharge well03:17:27
the remainders modulo k will also add up03:17:30
it turns out that the person is comparable to the correspondent, that is03:17:33
which is divisible by k03:17:37
now what are these numbers?03:17:42
what numbers you need to take were there03:17:44
some options but in my opinion03:17:47
for example, the game writes some garbage to03:17:52
unfortunately03:17:53
Power 10 k equals 1 that's not true03:17:57
not true03:17:59
10 to the power m03:18:02
walrus emma only ma03:18:07
that's true of course of course of course03:18:11
look at these friends let's go03:18:14
consider the number 10 to the pirate power03:18:19
Let's take the first number like this, I say03:18:22
that it gives remainder 1 modulo k03:18:27
Well, obviously an Euler creme03:18:36
trailer it's true03:18:39
this number is the sum of you paravyan and she03:18:42
does leave a remainder of 1 when divided03:18:44
on to03:18:46
cool03:18:48
but you need a lot of such numbers03:18:52
why can't I get by with just one thing?03:18:55
that when I start folding it myself with03:18:56
at some point the sum of the numbers is03:19:01
will go somewhere wrong, she will seem to take03:19:04
And03:19:06
the next digit will be carried over03:19:08
Here03:19:12
because we will do we03:19:14
we'll take it03:19:17
the following numbers are 10 to the power of 203:19:22
it is also comparable to the units I remember to the NEP03:19:25
square raised to03:19:29
10 to the power of 3 and from to you can also take and03:19:34
etc03:19:36
I can take as many such numbers as I like03:19:41
I'll take as much as I want03:19:47
Well, look, if I solved the problem for03:19:51
but not for n however I would like to receive03:19:54
sum of numbers to what I would do I would just03:19:57
added up like these numbers but I need03:19:59
get the sum of the digits m by03:20:02
how many numbers do you need to take to03:20:04
the sum of the numbers is equal to n03:20:09
also a question for you03:20:11
how many numbers like this should be taken to03:20:14
what date03:20:15
so that the sum of the digits is equal to i n03:20:21
that is, two 10s to the power n03:20:26
mother hen03:20:29
this is also comparable to units modulo03:20:33
k&n because I won't start from scratch Igor03:20:36
because the so-called03:20:42
now look what's funny about it03:20:45
these numbers it is obvious that03:20:49
a unit stands on different levels03:20:53
Right03:20:56
one in different categories, that is, such03:20:58
I add up the numbers let's admit it03:21:00
to myself03:21:02
These are some numbers I add up03:21:06
Eddie is being announced gradually03:21:10
when I add numbers I get a sum03:21:12
the numbers are also added to me03:21:15
transition through the discharge does not occur03:21:18
1001 but zero ends with zero03:21:21
that is, I will get some number03:21:25
1 bunch of zeros one bunch of zeros one bunch03:21:30
zeros and03:21:32
etc03:21:36
a bunch of zeros at the end03:21:41
which are divisible by n03:21:49
no n-na to03:21:51
until I got a number that is divisible by03:21:54
why else if I add m such numbers03:22:00
that is, the answer is from k to n of the drink then y03:22:05
I'll succeed03:22:11
the amount that is divided by03:22:14
but when I put it on and I can do it03:22:17
do with me03:22:18
m units will add up03:22:23
n is divided by kada o not yet excellent03:22:28
nice to see that the guys have a face too03:22:30
came03:22:31
Great03:22:34
no way to take badly because we want03:22:36
the sum n-well of Euler's theorem is of course 1903:22:40
no small company this is the maximum this is03:22:43
of course, like only if you are yourself03:22:46
want to apply it03:22:53
wow, this is not without proof thermite03:22:58
Hello03:23:00
so we understand, yes, that is, we will take ale03:23:03
numbers just so that the sum of the numbers is03:23:05
equal to n03:23:07
fog equals n03:23:14
so for some reason it’s obvious I’m still03:23:17
achieved only divisibility by k divisibility03:23:20
kicking I haven't received it yet they want the sum of the numbers03:23:22
equals n and is divisible by kana03:23:26
left to do03:23:31
So03:23:32
Nick the third one doesn't need ombrello mix then03:23:36
well, that is, if it’s difficult and unclear03:23:38
the third round doesn't have any more03:23:40
attitude towards 19 for honestly I will say yes to03:23:44
for the second one it's just a little bit a little bit03:23:47
maybe yes, at least there for two numbers03:23:49
a little bit and understanding are useful03:23:51
here we multiply by 1003:23:55
Well, here's 0 and add to the hook on write03:23:58
trailing zeros03:24:03
Here03:24:06
putting pressure on03:24:09
alpha plus b drowned03:24:13
albeto is definitely enough if the number03:24:16
will end in alt + b drowned it03:24:18
there will be 20 fire alpha and 20 503:24:20
degree beta share03:24:22
everyone got it03:24:25
security question guys why am I you no03:24:30
first 2 to the alpha power to the 5th power03:24:32
beta why I couldn’t immediately reason03:24:35
module n what for is it all module k03:24:38
why couldn't I just modulo n03:24:41
reason03:24:47
well done to Grigory03:24:52
because there may be 10 tons of oil on03:24:56
simple ones03:24:58
because I deliberately got a deuce03:25:02
gave out the five leaving as who03:25:05
reciprocally just with03:25:09
ten and only then can I apply03:25:13
Euler's theorem03:25:14
Of course it’s not possible without this03:25:18
Here03:25:20
Let's have a quick run one more time03:25:23
according to the proof it means firstly that03:25:27
we do but globally globally we03:25:30
we want to write a lot of numbers whose03:25:33
the sum of the digits will be equal to one and03:25:36
remainder when divided by k this is this k03:25:39
all that was left when we took out the deuce03:25:41
and the five was also the paradise of the one03:25:45
then I will add these numbers to you03:25:47
me and the sum of the numbers will add up and03:25:50
the remainders if I take n such numbers and03:25:53
I'll add it up03:25:54
then the remainder will add up and you will get the remainder03:25:58
w-well, a comparable son, that is, who03:26:02
is divided by k correctly here and on the left y03:26:05
I'll succeed, well in my case03:26:08
dozens of degrees of her buckwheat will be in different03:26:12
discharges when they add up there will not be03:26:14
no transitions through the discharge, that is03:26:16
it’s just that everything just adds up stupidly to the sum of the numbers03:26:18
will be equal to n03:26:20
Well, then we’ll struggle with divisibility by 203:26:24
constrained by 5 degrees beta uncomplicated03:26:26
I’ll just add how many zeros at the end03:26:28
necessary03:26:29
so I get divisibility by a power of two03:26:32
power of five03:26:35
By the way, you can write a plus sign if03:26:37
understood, I think that the task of this03:26:40
deserves from where well what from03:26:43
international olympiad if it's already03:26:46
the task of these is from some international03:26:47
sources but the fact itself is very beautiful03:26:51
quite simple if you do it right away03:26:54
evidence, no matter what the hell listen03:26:57
you'll come up with something03:26:58
it's really hard to come up with it honestly03:27:01
let's say03:27:03
I also need to come up with this idea03:27:10
so great great century Sasha super03:27:16
oh well03:27:19
give one last thing to the newcomer and to his mind03:27:23
ram targets03:27:26
task number 8 from this fox. he's on it03:27:31
Why is it possible to learn this with practice?03:27:33
invent03:27:37
Lisa is lost and they don’t look at us03:27:40
clear, clear03:27:43
Maybe03:27:46
counts the limits03:27:55
no pharma I know where Lisa is better than us03:27:57
You now have 7 children from servant 403:28:00
class just after one o'clock03:28:05
days from services and children03:28:09
4 interesting class03:28:17
Well, here's the last problem03:28:22
2 to the power 2 to the power of two and so on03:28:27
powers 2n03:28:31
deuces03:28:38
-2 to the power 2 to the power 2n03:28:43
-1 deuce03:28:52
shares03:28:54
for all numbers from03:28:57
one to n03:29:08
also quite simple polishing may not03:29:12
there was such a thing as task 503:29:14
but in principle everything is clear before the control03:29:18
By the way, the question is what is the difference between this task03:29:21
why can’t I immediately write share03:29:24
by n factorial03:29:27
why invent all this, why not03:29:29
write simply divided by n factorial03:29:51
goals03:29:53
What do you have in mind03:29:57
yes yes well in general it’s like03:30:00
there he really shares about the bike03:30:04
on and factorial means that03:30:06
divided by on hover but damn like03:30:09
number03:30:10
don't know 12003:30:15
bad number 60 is divisible by each of03:30:19
numbers 1 2 3 4 5 6 but on and factorial 103:30:24
a week by 6 factorial but not divisible03:30:26
it's just 6 factorial03:30:28
12003:30:30
Here03:30:34
Not now03:30:37
already 120 720 and 72003:30:42
that's why the wording is exactly03:30:45
this is divisible by all numbers from one to03:30:46
I03:30:49
Well, here I’ll immediately warn the guys that03:30:52
Of course, we will not solve this problem03:30:54
without induction03:30:56
Of course, not without induction now I03:30:58
I'll try on my fingers03:31:00
explain yes, basically, you don’t know03:31:03
what it is03:31:05
then, okay, maybe you won’t understand the problem03:31:09
it’s a real disaster, that is, for further03:31:13
tasks for Chinese prison remains03:31:15
for example it won't happen but right away03:31:17
I warn you03:31:18
although I’ll explain with my fingers what’s happening03:31:21
question if the meaning of my further if03:31:23
now you'll have problems with understanding03:31:26
computer physics and technology well, I would recommend03:31:29
see part of the rental now03:31:31
you can do at least half an hour there03:31:34
Our synod will have a break soon03:31:37
Here03:31:42
so on me for specifically in the naked03:31:46
understand what it is like03:31:48
arranged, leave a little bit, maybe03:31:50
a little useful03:31:52
Here03:31:55
please don't forget to put emoticons03:31:59
don't forget don't forget and03:32:01
so look, it means jokes are decided by03:32:05
induction03:32:07
planet ft murad a03:32:11
hear by induction, let's say03:32:15
let's check that 2 squared minus 203:32:17
divisible by all numbers from one to two03:32:20
yes deuces03:32:24
well obviously03:32:29
two is divided into one and two03:32:32
we will continue to do the following03:32:37
consider the number will collapse the expression03:32:41
where and there and there one deuce more but03:32:45
use what is for less03:32:47
the number of twos we have already proved everything03:32:53
Now we want to prove this, but03:32:56
we will do this gradually step by step03:32:59
that is, first we prove that 2 in 22 is minus03:33:04
2 square is divided by 1 2 3 and so on and03:33:08
this base was this base this base here03:33:14
further03:33:16
I won't check it anymore, there's no need to03:33:20
now be transition03:33:23
well it turns out that I want to prove yes03:33:26
this is the statement to prove for03:33:28
less number of twos I already know everything03:33:34
here is a quotation book yes yes of course I am03:33:39
I can't go to everyone03:33:43
So03:33:47
but look ok let me do it03:33:50
I’ll write the expression now, we’ll use it03:33:52
transform a little03:34:01
I want to do this first thing03:34:05
transformation comes to mind03:34:08
m03:34:10
well, you just need a natural number03:34:13
for his natural m this thing03:34:17
divided into all purity into water and03:34:19
I want to endure it until I endure it, but here it is03:34:22
the smaller number you let's put 2 in03:34:25
powers of two and so on 203:34:28
number in which03:34:31
n minus 1 two03:34:39
what will remain here03:34:45
how we endure let's realize03:34:51
Well03:34:54
since 2 to the power alpha -2 to the power b03:34:58
then when taking 2 to the power b it is03:35:00
turns into 2 to the power of alpha minus03:35:03
beta03:35:05
-1 don't forget, but just in case03:35:10
it works we just have to write03:35:11
the difference between these numbers03:35:18
about the square, be careful guys with03:35:20
neater squares, I hope you03:35:22
can you guess what03:35:25
ok let's do it since I have one03:35:28
Question: what is 2 to the power of 2?03:35:31
degree 203:35:38
plan but y is equal to 20 or 2203:35:44
degree 203:35:45
there 16 yes this is it03:35:54
256 guys say03:35:59
75603:36:03
no the definition of course does not depend03:36:07
absolutely here03:36:09
same procedure03:36:12
spent two 1603:36:15
his03:36:17
this thing03:36:19
little girl is always counted from top to bottom03:36:23
always top to bottom why03:36:26
because it's not 2 to the power of 2 by 2 by03:36:29
203:36:31
we now essentially calculated this03:36:34
the number is correct but I'm to the degree03:36:36
erected03:36:38
That's why I didn't leave parentheses here03:36:43
always count from top to bottom03:36:47
in this it is 2 to the power of 203:36:50
4 is correct03:36:53
2 1603:36:56
55000 how much is it I'm not one of these so03:37:00
I can't remember what it's like03:37:07
I understand guys03:37:12
considering that half the name for some reason03:37:14
first wrote 2 56 I think you already03:37:18
like something useful for yourself for sure03:37:22
carried out03:37:25
So03:37:30
not just one03:37:33
Pepe it’s not useful what to write03:37:38
such an expression in the Olympics can generally03:37:40
you simply think he’s wrong03:37:42
then just immediately pass by immediately meme here03:37:48
from above we must write down the difference here03:37:50
these numbers are correct03:37:57
here n minus 1 two03:38:02
here n minus 2 courtyards and03:38:07
minus 103:38:14
there you go03:38:18
I'm yours I haven't seen anyone03:38:24
so I don’t know anything, but we understand further03:38:27
this is how it goes03:38:32
Well, I just wrote Olka minutes03:38:37
just because alpha minus b to the03:38:42
volt fb2 brushes are one less than03:38:45
turrets taking into account this bottom two03:38:48
so here m minus 1 here and -2 here n03:38:52
minus 1 here n minus 203:38:59
there are minus ones here because I do it myself03:39:01
whole number down03:39:05
that's good03:39:09
look we got it cool03:39:12
and the number is this03:39:17
the same expression is not from the previous step03:39:20
induction03:39:22
what what exactly what can be endured like that03:39:25
that in the sense of degree he thinks so of course03:39:27
always the truth, always counting from the top03:39:30
top down03:39:34
from the task03:39:37
no no works with any03:39:43
we will mix it and understand that it will be conveyed03:39:46
works with any numbers03:39:56
you know we got a number with03:39:59
previous step by assumption03:40:02
I already know by induction that is divisible by03:40:04
all numbers from one to n minus 103:40:11
is already sharing03:40:15
on03:40:17
his he 1 to n minus 103:40:27
why exactly like that but look03:40:29
let's call it alpha03:40:32
let's call it beta and then I'll take out 20 pepe03:40:35
not beta03:40:37
this is a beautiful number k by 2 in03:40:41
degrees minus b then minus 1 numbers alpha and03:40:45
beta is one deuce less than well in03:40:48
initial for obvious reasons we just03:40:51
we forgot this deuce, we were crossing the alps03:40:54
we designated this [ __ ] as beta03:40:57
this means Wolf n minus 1 deuce in03:41:01
beta n minus 2 twos03:41:07
no it's not a square03:41:09
no no no no no03:41:10
this is exactly what the square guys are03:41:14
this [ __ ] is not a square, well03:41:17
there is not just a square prograf a square03:41:20
not this way03:41:22
if only there was a product of these03:41:24
two two just works then03:41:26
that would be squared three times03:41:29
A so no03:41:37
yes and -1 why because we are gradually03:41:41
we decide, look, I moved from one03:41:43
two two three three four03:41:46
now I'm solving a problem when I have03:41:48
first number of twos03:41:50
we get a problem where there was m minus 103:41:53
deuce 10 minutes two deuces I have already decided03:41:56
Well, I'm kind of gradually moving03:42:00
well it turns out03:42:02
so I know yes no it means known by03:42:07
assumption of induction03:42:16
Pavla rams will help03:42:19
how will it help me that the degrees are divided into03:42:22
all numbers from one to -1 are generally for me03:42:25
we must prove that the difference is divisible by k03:42:29
where k is from 1 year and03:42:38
how can it not be that the degree is shared by everyone03:42:41
numbers from one to n minus 103:42:48
let's not get distracted about this03:42:50
no need for ancient Rus's no need03:42:53
probably the topic should help or yes03:43:02
that is, I want to prove that 2 to the power03:43:06
this one here03:43:16
minus 1 two minus 2 powers 2 is03:43:19
drove 2 n minus 2 deuces03:43:23
comparable to unity modulo k03:43:32
Well, it's not always true03:43:37
if the chain mail is really like you03:43:40
rightly noted03:43:41
Well, it will be [ __ ] with even numbers to it03:43:44
it will work so let's probably go to03:43:47
since let's imagine03:43:50
but let k be a power of two03:43:54
2 to the power x times everything else03:43:59
we'll call everything else m03:44:05
Here03:44:08
degree two and I can't claim03:44:11
for divisibility but I can probably do it for him03:44:16
then we'll figure out the degree somehow03:44:18
two03:44:22
why is this below done?03:44:24
comparison which I will now circle03:44:27
why is this being done03:44:34
now let's get into it, a little bit left03:44:37
no and not to die nor to die we live until03:44:40
end03:44:41
lived to the end died like the hero of the nose did not die03:44:45
no no don't die or die forgot not03:44:47
die03:44:49
why is this [ __ ] being done?03:44:53
yes we are03:44:56
we know that deuce and m are already mutual03:44:59
simple means you can apply the theorem03:45:02
Euler that is, if this is this thing03:45:04
divided by m from above03:45:08
by 503:45:14
If03:45:16
it is divisible by and from m03:45:22
then victory03:45:32
well, it’s obviously less than 5 m for us03:45:37
n03:45:38
something m03:45:41
m less than or equal to03:45:44
some number from 1 to n I read and from03:45:49
this number03:45:50
Well, if only it is extremely equal to one03:45:53
tapped numbers are always less than n03:45:58
that is, from this number but less or03:46:02
equals n minus 103:46:11
I will write here so often03:46:15
but I m03:46:19
less than n this is obvious yes Emma herself03:46:23
less than or equal to n.a. when I03:46:26
I count and at least by one the number03:46:28
decreases03:46:30
Well, at least I don’t take into account the number m itself03:46:32
among numbers that are relatively prime prices03:46:36
except in cases equal to one but by03:46:38
unit of divisibility somehow03:46:40
obviously03:46:43
Means03:46:47
I m03:46:50
takes values from 1 to n minus 103:47:02
Well03:47:05
from 1 to n minus 103:47:09
Well03:47:12
All03:47:13
this thing is divisible by every number from 103:47:16
to n minus 1 means on fiat m too03:47:19
shares03:47:30
it's true she still has friends left03:47:33
this is the degree of milking, we have forgotten about it for now03:47:36
what is done with 2 to the power of x why with 203:47:39
to the power of x this thing will divide03:47:44
what expression should be divisible by 2 in03:47:46
degree x let's realize03:47:56
well, probably this thing will be on03:47:59
one sheet pin deuces share damn m03:48:03
minus 1 deuce03:48:05
it is obvious that by 2 to the power n she03:48:07
share03:48:10
there are a lot of deuces here, as if yes03:48:14
not one but not 2 to the power n minus 103:48:17
two is actually more03:48:19
it's not 2 to the power of two by two anymore03:48:22
and so on n minus 1 time is even more03:48:25
I don't just square it every time03:48:28
and I'm counting the degree03:48:31
that is, this thing is something that anyone will love03:48:34
by divisible by 2 to the x power03:48:39
Let's go through the logic one more time, well03:48:44
that I feel that I’m already so on the brink03:48:45
on the verge of this is the last task on the topicDescription:
Материалы для вебинара: https://1.shkolkovo.online/public-storage/f030b536-0907-4c9c-9c5c-0c22227c3dc6 Выложили крутейший КОНСПЕКТ, в материалах по ссылке выше!!! Хочешь готовиться к олимпиадной математике вместе с ДА и МО? Пиши ОЛИМП21 в сообщения сообщества https://vk.me/shkolkovo_ege ! Залетай на курс ещё и по скидке! Твой промокод на скидку 30% OLIMP30 ! Не забывай о подготовке к ЕГЭ! Узнай все подробности по ключевому слову ОСЕНЬ https://vk.me/shkolkovo_math Подписывайся на рассылку полезных материалов по математике: https://vk.com/app5898182_-185634090#s=950854&force=1 Все наши каналы на Ютубе: https://vk.com/wall-118664176_45573 Тайм-коды!! 0:00 Начало! 7:35 Остатки, сравнения по модулю и их свойства (17:40) 34:45 Разбор нескольких задач на сравнения по модулю 1)34:45 2a) 44:05 2b)45:05 2c) 46:35 3)55:30 1:01:15 Полная система вычетов (ПСВ) и ее свойства (1:09:15) 1:22:18 Приведенная система вычетов (ПрСВ) и ее свойства для простых m (1:26:28) 1:36:10 Теорема Ферма для простых и ее доказательство 1:47:15 За что бан в вк? 1:55:38 Почему теорема Ферма не работает так же для составных чисел 1:59:30 Функция Эйлера фи(m) 2:03:13 Свойство ПрСВ для составных m 2:07:10 Теорем Эйлера и ее доказательство 2:18:49 Теорема Вильсона и ее доказательство (2:21:00) 2:43:50 Разбор нескольких задач на теорему Эйлера 2) 2:43:50 4) 2:59:25 5) 3:08:00 8) 3:28:25 4:15:15 Начало второго модуля 4:18:00 Китайская теорема об остатках для двух чисел (КТО) и ее доказательство (4:21:53) 4:40:20 Китайская теорема об остатках для нескольких чисел (КТО) и ее доказательство (4:43:40) 4:55:35 Разбор нескольких задач на КТО 3) 4:55:35 4) 5:03:25 2) 5:10:20 7) 5:15:40 8) 5:24:55 11) 5:38:40 6:22:00 Начало третьей части 6:23:25 Показатель и его свойства 6:36:30 Разбор нескольких задач на показатель 5) 6:37:00 4) 6:50:50 7) 7:04:25 8) 7:14:10 7:46:35 Вычеты 7:55:15 Сколько существует квадратичных вычетов? 8:01:45 Свойства вычетов 8:12:15 Символ Лежандра и его свойства 8:28:30 Разбор нескольких задач на вычеты 4) 8:28:30 5) 8:31:45 9) 8:37:35 9:20:25 Начало четвертой части 9:21:20 Первообразный корень 9:35:35 Существование первообразного корня по простому модулю 10:32:00 Анекдот про цирюльника 10:47:00 Сколько существует первообразных корней по модулю p? 10:55:50 Доказательство теоремы Вильсона с помощью первообразного корня 11:00:55 Кубические вычеты и прочие прочие 11:10:00 Разбор нескольких задач на первообразные корни 3) 11:10:25 6) 11:17:20 11:26:15 Рождественская теорема Ферма 11:52:36 Анекдот про цирюльника. Повторяем, закрепляем
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