Let $a,b$ be elements in a group $G$ such that $|a|=25$ and $|b|=49$. Prove that $G$ contains an element of order 35.
If $G$ is finite then we can say that $5,7 \mid G$ and hence by Cauchy's Theorem, we have elements $c,d$ in $G$ of orders 5 and 7, respectively. So we can take $cd$ to be the element of order 35. What about the case when $G$ is infinite?