 Finite quotients of the integers localized away from one prime

If $A_{\neg p}=\mathbb{Z}[q^{-1}:q\ne p]$, its finite quotients are exactly the cyclic $p$-groups $\mathbb{Z}/p^n\mathbb{Z}$. Divisibility by every $q\ne p$ excludes other prime torsion, while a quotient of exponent $p^n$ factors through $A_{\neg p}/p^n{A}_{\neg p}\cong \mathbb{Z}/p^n\mathbb{Z}$.