Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 2 / c / ii / Solution

If $\pi$ omits a prime $p$, then $A_{\pi }\subseteq A_{\neg p}\subseteq {\mathbb{Z}}_p$ by part i. The reductionsseparate its nonzero elements, so their restrictions separate the elements of $A_{\pi }$. Thus $A_{\pi }$ is residually finite. If $\pi$ contains every prime, then $A_{\pi }=\mathbb{Q}$, which has no nontrivial finite quotient because it is a divisible group.