I would to have this formula and a demonstration of it.
2026-03-29 20:51:57.1774817517
There's a formula for the order of the elements in $\mathbb{Z}/(n)\times \mathbb{Z}/(m)$?
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$g\in\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\Leftrightarrow g=(g_m, g_n)$, where $g_m\in\mathbb{Z}/m\mathbb{Z}, g_n\in\mathbb{Z}/n\mathbb{Z}$, so one immediately has $g^{|g_m||g_n|}=(1, 1)$. It is not hard to check that $|g|=\text{lcm}(|g_m|, |g_n|)$.