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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.

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    $\alpha,\beta,n,m,r$ are all given?2012-03-01
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    Yes, they are all given, and $k$ is the indeterminate.2012-03-01

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