 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $a$ and $b$ generate $C_m$ and $C_n$. The order of $(a,b)$ in the <direct product of groups> is
$$
\operatorname{lcm}(m,n).
$$
When $\gcd(m,n)=1$, this is $mn=|C_m\times C_n|$. Hence $(a,b)$ generates the whole product and
$$
\boxed{C_m\times C_n\cong C_{mn}}.
$$

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