Prove that $n\mid\phi(a^n-1)$ in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses $\operatorname{Aut}(G)$?

Question

I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.
The following problem is Problem 16 on p.71 in Herstein's book.

Let $\phi(n)$ be the Euler $\phi$-function. If $a>1$ is an integer, prove that $n\mid\phi(a^n-1)$

This problem is for Section 2.8 "Automorphisms".
So, I think the author's solution for this problem uses $\operatorname{Aut}(G)$ for some group $G$.
But the natural solution below doesn't use $\operatorname{Aut}(G)$ for some group $G$.

https://math.stackexchange.com/a/398227/384082

Please tell me a natural solution that uses $\operatorname{Aut}(G)$ for some group $G$.

Answer 1

Note that $\phi(a^n - 1)$ measures the number of automorphisms of $\mathbb{Z}/(a^n - 1)\mathbb{Z}.$ There is a subgroup of order $n$ in this group: if $\phi$ is the automorphism sending $1$ to $a,$ then $\phi$ generates a subgroup of order $n.$ The statement follows from Lagrange's Theorem.