TSTST 2021 Problem 9

Let $q=p^{r}$ for a prime number $p$ and positive integer $r$. Let $\zeta=e^{\frac{2 \pi i}{q}}$. Find the least positive integer $n$ such that $$\sum_{\substack{1 \leq k \leq q \\ \operatorname{gcd}(k, p)=1}} \frac{1}{\left(1-\zeta^{k}\right)^{n}}$$ is not an integer. (The sum is over all $1 \leq k \leq q$ with $p$ not dividing $k$.)