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Math Help - congruence of binomial coefficients

  1. #1
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    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.

    Thank's in advance
    Last edited by hedi; March 9th 2014 at 04:23 PM.
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  2. #2
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    Re: congruence of binomial coefficients

    not touching a zip file sorry, post the image if you like
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  3. #3
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    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.
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  4. #4
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    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.
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  5. #5
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    Re: congruence of binomial coefficients

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