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