I am interested in the following related questions in metacyclic groups of the form $\mathbb{Z}_n \ltimes_r \mathbb{Z}_m$, where $r^n \equiv 1 \pmod{m}$:

The order of an arbitrary element $g = (\alpha, 0)*(0, \beta)$ - or some upper bound on the order - where * is the group operation.

The exponent of the group

I know that the first question reduces to finding the smallest integer $k$ such that:

$k \alpha \equiv 0\pmod{n}$, and

$\beta \frac{r^{k \alpha} - 1}{r^\alpha - 1} \equiv 0 \pmod{m}$,

but that's about it. Thank you very much in advance.