Proof of the following property of the totient function

Question

I am a newbie at number theory and was recently introduced to Euler's Totient Function. As I found it interesting, I decided to explore some of its properties. There are many of them on the Wikipedia page. I was able to the prove the following properties after seeing the proof of the multiplicative nature of the function. $$ \varphi(p_{1}^{r_{1}}p_{2}^{r_{2}}....p_{k}^{r_{k}})=(p_{1}-1)p_1^{r_1-1}(p_2-1)p_{2}^{r_2-1}.....(p_k-1)p_k^{r_k-1}$$ for $p_1,p_2....p_k$ being prime numbers. Another one I was able to prove was $$ \varphi(n) = n \prod_{p | n}\left( 1 - \frac{1}{p} \right)$$ However I am having trouble proving the following $$ n | \varphi(a^n -1) $$ for all $ a,n > 1$

I have no clue as to how to start this proof so any guidance will be highly appreciated.

FYI, this question is asked & answered at Prove that : $n \mid \varphi (a^{n}-1)$ $a>1$, but it involves more abstract algebra (e.g., groups and rings), than I suspect is appropriate for answering you. Also, there's a specific case of $a = 2$ at Euler's totient function Proof that $n\mid \varphi(2^n-1)$. — John Omielan, Oct 12 '21 at 06:35
Does this answer your question? $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer. — John Omielan, Oct 12 '21 at 06:41
It does but i guess i need to dive deeper into Group Theory to understand it. Thank you for your help! — VBM_375, Oct 12 '21 at 07:16

Proof of the following property of the totient function

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