Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 4 / b

By the Fundamental theorem of finitely generated abelian groups,with $T$ finite. If a nonzero element has a nonzero component in $T$, projection to $T$ separates it. Otherwise some integer coordinate is a nonzero $n$; choose a prime $p$ not dividing $n$ and reduce that coordinate modulo $p$. This gives a finite quotient in which the element survives, so every finitely generated abelian group is residually finite.