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From $ a, n \in \mathbb {N} $ with $ a> 1 $, show that $ n | \phi (a ^ n-1) $ where $\phi $ is Euler's phi function

Attemp: The way I was going about answering this question was considering $n-x$, where $n - x$ (mod $n$) $\equiv - x$ (mod $n$), and as gcd($-x,n$) = 1

trombho
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  • See https://en.wikipedia.org/wiki/Euler%27s_theorem. Also, this only holds when $\gcd(a,n)=1$. As a counterexample, you can just take $a=n$. For an example that feels a little less trivial, try $a=2$ and $n=10$. – Nikhil Sahoo Dec 06 '19 at 20:55

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