I have found proof of this here https://www2.math.ethz.ch/education/bachelor/lectures/fs2015/math/alg_topo/sol2.pdf but I find it very difficult to follow
I am trying to prove this for myself but I have many doubts, this is my attempt:
We know that $\operatorname{Tor}(A,B)=\operatorname{Tor}(T(A),T(B))$ where $T(A)$ and $T(B)$ are the subgroup of $A$ and $B$ respectively consisting of the torsion elements of $A$ and $B$. I could assume that there are $n\in \mathbb{N}$ and $m\in\mathbb{N}$ such that $T(A)\cong \mathbb{Z}/n\mathbb{Z}$ and $T(B)\cong \mathbb{Z}/m\mathbb{Z}$? In this case, conclude that $\operatorname{Tor}(A,B)\cong\mathbb{Z}/(m,n)\mathbb{Z}$ where $(m,n)=\gcd(m,n)$? Thank you.
Edit: this is a part of Exercise 6 of Hatcher's topology book page $267$