Fermat's Little Theorem/Corollary 1/Proof 1

Corollary to Fermat's Little Theorem

If $p$ is a prime number, then $n^p \equiv n \pmod p$.

Proof

There are two cases:

$(1): \quad$ If $p \divides n$, then $n^p \equiv 0 \equiv n \pmod p$.

$(2): \quad$ Otherwise, $p \nmid n$.

Then, by Fermat's Little Theorem, $n^{p-1} \equiv 1 \pmod p$.

Multiplying both sides by $n$, then by Congruence of Product we have:

$n^p \equiv n \pmod p$

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences: Theorem $3$: Corollary