SIMON-MARAIS 2022 A4

Let $n$ be a positive integer, and let $q \geq 3$ be an odd integer such that every prime factor of $q$ is larger than $n$ . Prove that

$\frac{1}{n! (q-1)^n} \prod_{i=1}^{n} (q^i - 1)$

is an integer that has no prime factor in common with $\frac{q-1}{2}$ .

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