TITLE: If $p$ is a prime number,show that $2(p-3)!+1$ is a multiple of $p$.
QUESTION [5 upvotes]: I have got a question which is as follows:

If $p$ is a prime number, show that $2(p-3)!+1$ is a multiple of $p$.

I know that this question can be solved using Wilson theorem which is the only thing I apply in such situations, but I don't know how can I use it here.
Please help.

REPLY [3 votes]: $(p-1)! \equiv -1 \mod p$ (Wilson's theorem). So, $(p-3)!(p-2)(p-1) \equiv (p-3)!(-2)(-1) \equiv 2*(p-3)! \equiv -1 \mod p$. So, $2*(p-3)!+1 \equiv 0 \mod p$