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 sum of $G$ and $G'$ amalgamated by their common subgroup $\mathbf{Z}$. Show that $G$ and $G'$ are residually finite, but that $E$ is not. (Prove that the only subgroup of $E$ of finite index is $E$.)