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Math Help - Binomial coefficient

  1. #1
    MHF Contributor alexmahone's Avatar
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    Binomial coefficient

    Let p\geq3 be a prime number, and let m and k<p^m be positive integers. Show that \dbinom{p^m}{k} is divisible by p.
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  2. #2
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    Re: Binomial coefficient

    Quote Originally Posted by alexmahone View Post
    Let p\geq3 be a prime number, and let m and k<p^m be positive integers. Show that \dbinom{p^m}{k} is divisible by p.
    We have the simple identity \binom{p^m}{k}=\binom{p^m-1}{k-1}\cdot\frac{p^m}{k} for 0<k. Hence, (a) k\cdot\binom{p^m}{k}=\binom{p^m-1}{k-1}\cdot p^m.

    Let p^j be the largest power of p that divides k. So \frac{k}{p^j}\cdot\binom{p^m}{k}=\binom{p^m-1}{k-1}\cdot p^{m-j}, by (a).

    From p^j\leq k<p^m it follows that m-j>0 and so p divides \frac{k}{p^j}\cdot\binom{p^m}{k}, where p does not divide \frac{k}{p^j}. Euclid's Lemma implies then that p must divide \binom{p^m}{k}.


    By the way, the argument works for p=2 also.
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