Given an abelian group $X$, there's a correspond $n$-torsion subgroup defined as follows: $X[n] = \{x \in X: nx = 0\}.$ Call a pair $(X,n)$ generative iff $\mathbb{Z}\langle X[n] \setminus X\rangle = X$, where the notation $\mathbb{Z}\langle S \rangle$ denotes the abelian subgroup generated by subset $S$.
Question. Which pairs $(X,n)$ are generative, where $X$ is a finitely-generated abelian group and $n$ is an integer?
Clearly, the answer is "not all of them." For example, $(\mathbb{Z}/2,2)$ does not have this property; but $(\mathbb{Z}/2,3)$ does. I was think of puzzling out the circumstances under which $(\mathbb{Z}/p^n,b)$ is generative, where $p$ is prime, and then trying to use the structure theorem to piece together the answer to the general problem. I don't have time to think about this carefully right now, but feel free to think about it if it seems interesting or cool :)