Number Theory: use Wilson's Theorem (p-1)!=-1 mod p to prove

[hcy780316]

Let n be a positive integer, and let p be a prime divisor of n!+!. Prove that p > n.

I use Wilson's Theorem (p-1)! = -1 mod p to prove

n!+1=0 mod p (According to the question)
so n! = -1 mod p

Can I use Wilson's Theorem (p-1)! = -1 mod p and n! = -1 mod p
to prove that n has be to be p-1 by any method?

 

[daon]

If p <= n then p|n!.

So p|n! => n! = p*r

And we're given p|(n!+1) => n!+1 = p*s => n! = p*s-1

So we have p*k=p*s-1, i.e. p*(s-k) = 1. No primes divide 1.