# Thread: Modulus proof involving least positive residues

1. ## Modulus proof involving least positive residues

I'm a bit stuck on this question:
Let p be a prime number. Prove that (2p−1)(2p−2)···(p + 1) ≡−1(mod p).
Hint: ﬁnd the least positive residues of 2p−1,2p−2,...,p + 1 (modulo p).
I think the least positive residues are {0,1,2,...,p} but not entirely sure, and I don't see how it's going to help me answer the question below. Thanks, Alex

2. ## Re: Modulus proof involving least positive residues

The least positive residues are 1,2,3,...,p-1 giving

$(p-1)! = -1$ (mod $p$)

Wilson's Theorem