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Explaining $\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(a^i-1\right)$ by a free $S_n$-action
Here is an olympiad-level problem on elementary number theory:
Let $a$ be an integer and $n$ a positive integer. Prove that
\begin{align}
\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(...
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