Simple division problem with factorial

Question

Assume $n \neq 1$ and $n \in N$. For what values of $a \in N\ a^{n!}-1$ can be divided by $n$?

It looks like I have to use an Euler division theorem, but I have no idea how to apply it here.

Answer 1

It obvious if $\gcd (a,n)>1$ then $n \nmid a^{n!}-1$. If $\gcd (a,n)=1$ then according to Euler's theorem $a^{\varphi(n)} \equiv 1 \pmod{n}$. But since $\varphi(n)<n$ so $\varphi (n) \mid n!$ implying $n \mid a^{\varphi(n)}-1 \mid a^{n!}-1$.

Thus, for all $a$ so that $\gcd (a,n)=1$ then $n \mid a^{n!}-1$.