# Thread: congruence of binomial coefficients

1. ## congruence of binomial coefficients

Hi,

I need help for proving :

let p be a prime, and a and b be any integers such
that a ≥ b ≥ 0. Prove that the binomial pa over pb is congruent modulu p to the binomial a over b.

2. ## Re: congruence of binomial coefficients

not touching a zip file sorry, post the image if you like

3. ## Re: congruence of binomial coefficients

Hi,
Here's the result you need: Legendre's Theorem - The Prime Factorization of Factorials. Just apply this to (pa)!, (pb)! and (p(a-b))! and it should be clear.

4. ## Re: congruence of binomial coefficients

Oops. Sorry, as far as I can tell, the previous post merely shows:

$$\nu_p{pa\choose pb}=\nu_p{a\choose b}$$

So one binomial is 0 mod p iff the other is also 0, but this does not solve the problem generally.

5. ## Re: congruence of binomial coefficients

Yes,this was my difficulty.It can be solved by the binomial expansions of to p-congruent binomials.

### binomial congruence

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