# prove if p is prime then...

• February 10th 2008, 05:44 PM
mandy123
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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• February 10th 2008, 07:10 PM
ThePerfectHacker
Quote:

Originally Posted by mandy123
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 $pk!$.