If $p, q$ are primes and $p$ does not divide $n-1$ and $p$ divides $n^q-1$. How to prove that $q|p-1$.
Let $p$,$\ q$ be prime numbers and $n\in \mathbb{N}$ such that $p \nmid n-1$. If $p \mid n^q-1$, then $q \mid p-1$.