Proof of Korselt criterion

If every integer satisfies $a^n\equiv a(\mathrm{mod}n)$, taking $a=p$ shows that $p^2$ cannot divide $n$. Taking a primitive root modulo each prime divisor $p$ shows $p-1\mid n-1$. Conversely, these divisibilities and squarefreeness let Fermat's little theorem prove the congruence modulo each prime divisor, and coprimality of those divisors combines it modulo $n$.