  We investigate the $K$-theory of unital UCT Kirchberg algebras
$\mathcal{Q}_S$ arising from families $S$ of relatively prime numbers. It is
shown that $K_*(\mathcal{Q}_S)$ is the direct sum of a free abelian group and a
torsion group, each of which is realized by another distinct $C^*$-algebra
naturally associated to $S$. The $C^*$-algebra representing the torsion part is
identified with a natural subalgebra $\mathcal{A}_S$ of $\mathcal{Q}_S$. For
the $K$-theory of $\mathcal{Q}_S$, the cardinality of $S$ determines the free
part and is also relevant for the torsion part, for which the greatest common
divisor $g_S$ of $\{p-1 : p \in S\}$ plays a central role as well. In the case
where $\lvert S \rvert \leq 2$ or $g_S=1$ we obtain a complete classification
for $\mathcal{Q}_S$. Our results support the conjecture that $\mathcal{A}_S$
coincides with $\otimes_{p \in S} \mathcal{O}_p$. This would lead to a complete
classification of $\mathcal{Q}_S$, and is related to a conjecture about
$k$-graphs.
