Finite generation detected by a prime quotient ($A\text{ finite}⟺A/pA=0\text{ for some prime }p$)

The Fundamental theorem of finitely generated abelian groups writes $A={\mathbb{Z}}^r\oplus F$ with $F$ finite. Reduction modulo a prime contains $(\mathbb{Z}/p\mathbb{Z}{)}^r$, so a vanishing quotient forces $r=0$. Conversely multiplication by a prime coprime to $|F|$ is bijective on $F$. Finite generation matters: the nonzero divisible group $(\mathbb{Q},+)$ satisfies $p\mathbb{Q}=\mathbb{Q}$ for every prime and is not finitely generated, since any finite list of rational generators has a common denominator.