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Math Help - Prime factors and binomial coefficients

  1. #1
    Junior Member Greg98's Avatar
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    Prime factors and binomial coefficients

    Hello,
    First sorry about plain ASCII (LaTeX isn't working atm :/). The task is to prove following theorem: if 1<k<p is prime factor of p, then C(p, k) isn't divisible by p.

    I think I have seen a proof, which states that: If p is a prime and 0<r<p, then C(p, r) is divisible by p. But, I don't remeber, how that was proven.

    So, any help is appreciated. Thanks in advance!
    Last edited by Greg98; April 26th 2011 at 11:33 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Greg98 View Post
    Hello,
    First sorry about plain ASCII (LaTeX isn't working atm :/). The task is to prove following theorem: if 1<k<p is prime factor of p, then C(p, k) isn't divisible by p.

    I think I have seen a proof, which states that: If p is a prime and 0<r<p, then C(p, r) is divisible by p. But, I don't remeber, how that was proven.

    So, any help is appreciated. Thanks in advance!
    How rigorous do you want to be? Write out C(p,k) and note there is a factor of p in the numerator and since all the factors in the denominator are strictly less than p none of them can 'cancel' the p.
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