a) Let $G = \langle a \rangle$ be a finite cyclic group. Prove that for each $b\in G$, $\langle b \rangle=G$ if and only if order of $b$ equals order of $G$.
b) The previous part does not hold if $G$ is an infinite cyclic group. Why not?
(a) How many elements can $b$ generate? How many elements are in $G$?
(b) What is the prototypical infinite cyclic group?
In future, please give some indication of what you've tried out, and what you are stuck on.