TITLE: $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
QUESTION [10 upvotes]: Let $n$ and $a$ be positive integers with $a > 1$. I need to show that $n$ divides $\phi(a^n -1)$.

Here, $\phi$ denotes the Euler totient function.
Could any one give me a hint?

REPLY [6 votes]: You can apply Euler's Theorem like this:
$${a}^{\phi\left(m^n-1\right)}\equiv 1 \pmod{m^n-1}$$
Also, you can use this fact:
$$\left(x^a-1,x^b-1\right)=x^{\left(a,b\right)}-1$$