Exercise 35, § 7

\* Let $G$ (resp. $G'$) be the additive group of fractions of the form $a/b$, with $a \in \mathbf{Z}$, $b \in \mathbf{Z}$ and $b$ not divisible by 2 (resp. by 3). Let $E = G *_{\mathbf{Z}} G'$ be the amalgamated sum of $G$ and $G'$ by their common subgroup $\mathbf{Z}$. Now prove that $G$ and $G'$ are residually finite, but that $E$ is not. (Prove that the only subgroup of finite index of $E$ is $E$.)*