Number Theory | Chinese Remainder Theorem Proof

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We present a proof of the Chinese Remainder Theorem.

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You mathematicians are so ingenious. I get the shivers out of seeing this proof. It is just beautiful. Thank you for posting this video.

lucasm.b.

Great video. Now it gets very clear to me, since I was struggling to understand the part of proof from wikipedia. In wikipedia, it is given 4 sections of proof:
Uniqueness,
Existence proof 1 (by proving the existence of a map from x mod N to (x mod n1, ..., x mod nk) that defines ring isomorphism),
Existence proof 2 (constructive proof only on 2 divisiors; n1 and n2; this part is that I get so confused on how they construct x from x = a1m2n2 + a2m1n1 for ai are the remainders, mi are Bezout's coefficients),
Existence proof 3 (the general proof as that of yours).
I found yours enlightening. Thanks Professor, keep creating more videos of mathematics, cheers 👍.

rizalpurnawan

Great video! Part about d seems a bit inaccurate because d can have non-one factors that are distributed over multiple nj (j n.e. i). Stating in advance that d is either 1 or a prime factor of ni should make it accurate.

harrysakata

I was just casually surfing Number Theory youtube and came across your tutorial. Stopping at 7:32 (obviously going to resume again) to just let you know how awesome this video is. (consider adding the alternative bit of solving the system through residue mod Pi) but all in all, you teach really really well!

devesh

I think you are doing a great service in these videos. A long time since school, you make the Olympiad and other similar challenging problems I could never solve understandable to me. The proof of this theorem reminded me of orthogonal polynomials and Gaussian integration, which might make a good topic, if you haven't covered it.

Don-evov

Professor M. Penn, thank you for a mammoth proof of the classical Chinese Remainder Theorem.

georgesadler

Hats off to your dedication to explaining mathematics, to your lung power and not the least to your amazing back flips.God bless you with more and more opportunity to serve the cause of Mathematics. Thank you.

prakashmadaksirashamrao

He finishes with "okay, good". Shivers.

danielkoynov

I always feel a bit of discomfort when quantifiers are being added at the end of the statement. Years of inculcation made it.

Suav

If d divides both n_i and n_j, then d is a common divisor of n_i and n_j. But gcd (n_i, n_j) = 1, so d=1. Simpler.

johnvandenberg

Is it true that you are a master at making pizza?

adenpower

I was amazed that this Chinese remainder theorem can be applied to prove that Euler's totient function is multiplicative under two relatively prime integers.

사기꾼진우야내가죽여

I need help at 3:19. We suposed that d|n_i and d|N_i because d is the gcd(n_i, N_i). We know that gcd(n_i, n_j)=1 so if w|n_i it cannot divide n_j for w not equal to 1. But then the prof concludes that d|n_j. Any help pls? Thank you

fonso-irxz

i have a question : in the proof of chinese theorem if n1 = 4, n2=3, n3 = 5, then 2 divide 4 and 4*3*5 but 2 doesn't divide 3 or 5 . but you then said that d(=2) divide nj for every j different of ni(=2 in this case)

sebastianocampo

I don't understand the last part, how does X ≡ Y mod N show uniqueness?

mysteriousperson

Thanks for your video! I am just wondering how why X can be constructed as What's the idea behind this? Thanks!

fredrickelvis

It's good but my request to you is, next time before you proceed to proof please give some examples and please don't wipe out the proof in between

escobarlohe

sorry but why are we considering x to be that sum of products?

lgooch

my dad gave me one day to understand this i have to watch this video

Timmy-zlgu

can't understand last step "=>"

opsschwarz

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