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Math Help - prove if p is prime then...

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    prove if p is prime then...

    Prove that if p is prime and 1 less than or equal to k less than p then the binomial coefficient p = (p!)/(k!(p-k)!) is divisible by p.
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    Quote Originally Posted by mandy123 View Post
    Prove that if p is prime and 1 less than or equal to k less than p then the binomial coefficient p = (p!)/(k!(p-k)!) is divisible by p.
    This proves that for any 1\leq k\leq p-1 we have that p(p-1)(p-2)...(p-k+1) is divisible by k!. Now \gcd(k!,p) = 1 because p is a prime this means (p-1)(p-2)...(p-k+1) is divisible by k! thus p(p-1)(p-2)...(p-k+1) is divisibly by pk!.
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