Fermat's little theorem in the form a^p = a

Statement

Let $p$ be a prime number and let $a$ be an integer. Then $a^p \equiv a \pmod p$.

Remarks

If $p \nmid a$, multiply Fermat's little theorem $a^{p-1} \equiv 1 \pmod p$ by $a$. If $p \mid a$, both sides are $0$ modulo $p$. Conversely, this form implies the form $a^{p-1} \equiv 1$: cancel $a$, which is allowed when $p \nmid a$. In the ring $\mathbb{Z}/p\mathbb{Z}$, the form says that $x^p = x$ for each element $x$.

Explanations

L1 human

The form $a^p \equiv a$ needs no hypothesis on $a$. For example, with $p = 3$: $0^3 = 0$, $1^3 = 1$, $2^3 = 8 \equiv 2$, and $4^3 = 64 \equiv 1 \equiv 4 \pmod 3$.