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August 25th, 2013, 03:36 PM   #1
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Factorial and primes

Hi,
Does anybody where can I find the proof of the following:

Let (n,k) be the standard combination (choose k among n). Then if

(n,1)=(n,2)=...=(n,n-1)=0 (mod p),

then n=p^t for a non-zero integer t, where p is a prime.

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August 29th, 2013, 09:10 AM   #2
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Re: Factorial and primes

It's fairly easy to prove the contrapositive (hence proving the original statement). Suppose that n has a factor other than p. Then would not be divisible by p, since the denominator would have the same number of factors of p as the numerator. And if n is less than p, then it is obvious that no combination of n objects would be divisible by p since p would not appear in the numerator or denominator.
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