Famous Theorems of Mathematics/Fermat's little theorem

Fermat's little theorem (not to be confused with [[../Fermat's last theorem/]]) states that if $p$ is a prime number, then for any integer $a$, $a^p-a$ will be evenly divisible by $p$. This can be expressed in the notation of modular arithmetic as follows:

$a^p\equiv a(\mathrm{mod}p).$

A variant of this theorem is stated in the following form: if $p$ is a prime and $a$ is an integer coprime to $p$, then $a^{p-1}-1$ will be evenly divisible by $p$. In the notation of modular arithmetic:

$a^{p-1}\equiv 1(\mathrm{mod}p).$