Prove that every subgroup of a cyclic group is cyclic. Do this by working with exponents, and use the description of the subgroups of $\mathbb{Z}^+$.
Having taken a look at a solution after having struggled with this problem, I'm very confused about the hint. There is no guarantee that $S$ be a subgroup of $\mathbb{Z}^+$, e.g., it could contain non-integers, so I don't know where I need to use the fact that all subgroups of the positive integers are of the form $b\mathbb{Z}$.
A hint on how to use this approach would be appreciated.
You already know that if $H$ is a subgroup of $\mathbb{Z}$, then $H=a\mathbb{Z}$ for some unique $a\ge 0$.
Want to prove: If $G$ is a cyclic group and $H$ is a subgroup of $G$, then $H$ is cyclic too.
Let $G$ be a cyclic group and let $H\le G$ (notation for subgroup). Can you find a a surjective homomorphism $f\colon\mathbb{Z}\to G$?
Then $H=f(f^{-1}(H))$ and, since $f^{-1}(H)=a\mathbb{Z}$ is cyclic, $H$ is also cyclic, since it's the homomorphic image of a cyclic group (prove this).