This question already has an answer here:
The Question is this:
How many generators are there of the group $G\times H$, if $G$ and $H$ are cyclic groups of order $m$ and $n$, which are coprime?
Let's say that $G$ is generated by $g$, and $H$ by $h$. Here I already proved that $(g,h)$ is generator of $G\times H$ but I can't come up with another one. So does $G\times H$ really have any other generator?