modular arithmetic - Fermat's Little Theorem for composite numbers

In Fermat's Little Theorem we have that for a prime number $p$, $a^p \equiv a\pmod{p}$, is this ever true when $p$ is not a prime number, i.e. take some $n\in\mathbb{Z}$, would it hold true that $a^n \equiv a\pmod{n}$? Obviously here we assume $\gcd(a,n)=1$.