I'm confused about the following problem from Problems in Algebraic Number Theory by Murty:
Show that $n \mid \varphi(a^n -1)$ for any $a > n$.
I suppose $n \in \mathbb{Z}_{>0}$. The proof is quite straightforward using group theory: since $a^n \equiv 1 \pmod{a^n - 1}$ and $n$ is the smallest positive integer with that property (if $a^n - 1 \mid a^k - 1$, necessarily $k \geq n$), we have $\mathrm{ord}_{a^n - 1}(a) = n$. Then Euler's theorem gives that $a^{\varphi(a^n -1)} \equiv 1 \pmod{a^n -1}$, and the result follows from Lagrange's theorem.
My question: where is the hypothesis $a > n $ used? Is it necessary?
