Consider the congruence $x^{n} \equiv 1 \pmod{p}$, $p$ prime. One of my friends mentioned that it suffices to suppose that $n$ divides $p - 1$ since if not the number of solutions to $x^{n} = 1$ is equal to the number of solutions to $x^{m} = 1$ where $m \mid p - 1$. I'm skeptical, why is this true?
Counting solutions to a congruence mod $p$
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elementary-number-theory