Fermat's little theorem

Modulo a prime $p$, every integer satisfies $a^p\equiv a$.Suggest a slogan

Let $p$ be a prime and let $a\in\ZZ$. Then \[ a^p\equiv a\pmod p, \] and if $p\nmid a$, then $a^{p-1}\equiv1\pmod p$.

If $p\nmid a$, the class of $a$ lies in the group $\FF_p^\times$ of order $p-1$, so $a^{p-1}\equiv1\pmod p$ by Lagrange’s theorem, and multiplying by $a$ gives $a^p\equiv a\pmod p$. If $p\divides a$, both sides of $a^p\equiv a$ are $0$ modulo $p$.