Notes - Groups TT23, Chinese remainder theorem

Can you state the chinese remainder theorem, in the context of cyclic groups?

The Chinese remainder theorem states that if $m$ and $n$ be coprime natural numbers, then $C _ m \times C _ n \cong C _ {mn}$. Specifically, if $g$ is a generator for $C _ m$ and $h$ is a generator of $C _ n$ then what is a generator of $C _ {mn}$?

The chinese remainder theorem states that if $m$ and $n$ be coprime natural numbers, then $C _ m \times C _ n \cong C _ {mn}$. When proving this, what do you aim to show about the order of $(g, h)$ (say $k$) if $g$ is a generator for $C _ m$ and $h$ is a generator for $C _ n$?

Prove the chinese remainder theorem for cyclic groups:

Let $m$ and $n$ be coprime natural numbers, then $C _ m \times C _ n \cong C _ {mn}$. Specifically, if $g$ is a generator for $C _ m$ and $h$ is a generator of $C _ n$ then $(g, h)$ is a generator of $C _ {mn}$.