TITLE: If $p$ is prime, then $n\mid\varphi(p^n-1)$
QUESTION [5 upvotes]: How can I prove that the value of $\varphi(p^n-1)$ (where $p$ is prime and $n$ is some positive integer) is some multiple of $n$? The purpose of this is to prove that $n$ divides $\varphi(p^n-1)$.

REPLY [6 votes]: Consider the group of units $\Bbb Z/D\Bbb Z^\times$ where $D=p^n-1$, $p$ a prime.
What is the order of $p$?