Prove that:
$$11! + 1$$ is a prime number. Without computing the number (or factorial).
Obviously, from Wilson's theorem, a number $n$ is prime if,
$$(n-1)! + 1 \equiv 0 \pmod{n}$$
Since $n = 11! + 1 \in \mathbb{N}$, it is prime iff
$$(11!)! + 1 \equiv 0 \pmod{11! + 1}$$
I have a problem here, how do I use Wilson's theorem with factorials?
For a beginning,
Multiples of 11:
$$11, 22$$
$11! = 11*10*9...2*1 = 22*10!$
Next,
$$(11!)! = (22*10!)! $$
I need help at this point..