Let $G_{1}, G_{2}$ be groups such that $(|G_{1}|,|G_{2}|)=1$ (the $\mathrm{gcd}$). If $G_{1}, G_{2}$ are cyclic, show that $G_{1} \times G_{2}$ is a cyclic group as well (in particular, $\mathrm{ord}((g,h))=\mathrm{ord}(g)\mathrm{ord}(h)$).
I'm not sure on how to use the assumption to begin tackling the problem. I know that $\mathrm{ord}((g,h))=\mathrm{lcm}(\mathrm{ord}(g),\mathrm{ord}(h))$, so that could be useful. What would be the appropriate way to do this problem?
Thanks for the help.