I'm asked to prove the following statement:
Let $N=(1008!)^2+1$. Prove that $N$ is divisible by $2017$. (Hint: $2017$ is prime.)
I don't know how to go about proving this statement, since there seems to be nothing particularly special about this number except the factorial, which may point to some usage of Wilson's theorem. However, I don't know how to continue from there. Any hints? I would appreciate them a lot. (Please don't write full solutions, as I want to gain the intuition myself on how to solve these.)
Here's a hint:
You suggested using Wilson's theorem. However, that would require having a $2016!$ term. However, something about this problem suggests that a $2016!$ term may be hiding somewhere; you have something, $(1008!)^2$, with $2016$ factors. See if you can show that $$(1008!)^2\equiv 2016!\bmod 2017.$$