Fermat's Little Theorem/Corollary 4

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

Then:

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


Proof

By corollary 3 of Fermat's Little Theorem:

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

That is:

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

Since $p \nmid n$, by Corollary to Divisors of Product of Coprime Integers:

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

That is:

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

$\blacksquare$


Sources