Abstract Algebra Problem 1

Let $b$ be the element in $G'$ whose order is 8. As it is an epimorphism we know that there exist $a$ in $G$ such that $\phi(a)$ $= b$ .We know that order of $\phi(a)$ divides $o(a)$ .
Now as evident the order of the $a$ has to be a multiple of $8$ . So let order of $a = 8k$ where $k$ is some positive integer. Now as order of $a$ divides order of $G$ . Order of $G$ has to be of the form $8km$ . Where m is some other positive integer. Now order of $a^k$ would be 8 . Hence there exist an element of order 8 in $G$