Fermat's Little Theorem/Corollary 4

Corollary to Fermat's Little Theorem

Let $p^k$ be a prime power for some prime number $p$ and $k \in \Z_{\gt 0}$.

Let $n \in \Z_{>0}$ with $p \nmid n$.

Then:

$n^{p^k - 1} \equiv 1 \pmod p$

Proof

From Fermat's Little Theorem: Corollary $3$:

$n^{p^k} \equiv n \pmod p$

That is:

$p \divides \paren {n^{p^k} - n} = n \paren {n^{p^k - 1} - 1}$

Since $p \nmid n$, from Prime Divisor of Coprime Integers:

$p \divides \paren {n^{p^k - 1} - 1}$

That is:

$n^{p^k - 1} \equiv 1 \pmod p$

$\blacksquare$