# congruence of binomial coefficients

• Mar 9th 2014, 03:32 PM
hedi
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.

• Mar 9th 2014, 03:52 PM
romsek
Re: congruence of binomial coefficients
not touching a zip file sorry, post the image if you like
• Mar 10th 2014, 07:40 AM
johng
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.
• Mar 10th 2014, 09:15 AM
johng
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.
• Mar 10th 2014, 11:24 AM
hedi
Re: congruence of binomial coefficients
Yes,this was my difficulty.It can be solved by the binomial expansions of to p-congruent binomials.