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 May 13th, 2015, 04:46 AM #1 Member     Joined: May 2014 From: India Posts: 87 Thanks: 5 Math Focus: Abstract maths! Proof regarding divisibility of factorial by composite numbers. If N is a composite number greater than 4, prove that: $(N - 1)! \equiv 0(mod N)$
 May 13th, 2015, 04:50 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Let N = ab with a,b coprime and greater than 1. Then since a and b are both less than N-1, both divide (N-1)! and hence ab divides (N-1)!, as desired. All that's left is for you to handle the case of powers of primes.
May 13th, 2015, 05:16 AM   #3
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Quote:
 Originally Posted by CRGreathouse All that's left is for you to handle the case of powers of primes.
That's exactly where I am having a problem.

May 13th, 2015, 05:50 AM   #4
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Quote:
 Originally Posted by Rishabh That's exactly where I am having a problem.
It's a pity you didn't mention that, or I would have worked out that case instead of the other. Why don't you show me the work you've done so far and I'll see if I can help you along?

 May 13th, 2015, 11:36 PM #5 Member   Joined: Jul 2014 From: israel Posts: 76 Thanks: 3 this will help you out Wilson's theorem - Wikipedia, the free encyclopedia Wilson's theorem states that a natural number n > 1 is a prime number if and only if (N−1)!≡-1(mod N) Lagrange gave the first proof in 1771 please also check the "Composite modulus" section ... Last edited by isaac; May 13th, 2015 at 11:40 PM.

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