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:

$a^{p}\equiv a{\pmod {p}}.\,\!$

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:

$a^{p-1}\equiv 1{\pmod {p}}.\,\!$