Residually finite group with finitely many conjugacy classes of elements of finite order

Question

Let $G$ be a residually finite group. Show that if $G$ has finitely many conjugacy classes of elements of finite order then $G$ has a torsion free finite index subgroup.

Not sure how to get started on this one!

Answer 1

Hint: Start by constructing (finitely many) homomorphisms to finite groups which do not send representatives of conjugacy classes of finite order elements to 1. Now think about kernels of these homomorphisms.

Answer 2

Let $x_1, \cdots, x_r$ be the representives for these conjugacy classes of finite orders. Since $G$ is residually finite, there exists $N_i \lhd G$ of finite index in $G$ such that $x_i \not \in N_i$ for each $i$. Let $N=\cap N_i$, then $N$ is what you need.