# Fermat's Little Theorem **The difference between an integer and the integer to the power of a prime is a multiple of the prime.** > [!Example] Fermat's Little Theorem > If $p$ is a prime, then **Fermat's little theorem** states that for all $a\in\mathbb{Z}$, $a^{p}-a=kp,\quad k\in\mathbb{Z}$ > > In [[modular arithmetic]] notation, > $a^{p}\equiv a \pmod p$ > If $a$ is *coprime* to $p$, then $a^{p-1}-1$ is an integer multiple of $p$. That is, > $a\nmid p \implies a^{p-1}\equiv 1 \pmod p$