I am trying to solve the following equation system for an integer $k$: $\begin{align*} k \alpha &\equiv 0\pmod{n}\\ \beta \frac{r^{k \alpha} - 1}{r^\alpha - 1} &\equiv 0 \pmod{m} \end{align*}$ where $r^n \equiv 1 \pmod{m}$.
I could also use some bounds on $k$. This system arises when I try to determine the order of elements in a semidirect product of cyclic groups $\mathbb{Z}_n \ltimes_r \mathbb{Z}_n$. Thank you very much in advance.