We investigate the K-theory of unital UCT Kirchberg algebras QS arising from families S of relatively prime numbers. It is shown that K∗.QS/ 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 AS of QS. For the K-theory of QS, the cardinality of S determines the free part and is also relevant for the torsion part, for which the greatest common divisor gS of fp 1 V p 2 Sg plays a central role as well. In the case where S ≤ 2 or gS = 1 we obtain a complete classification for QS. Our results support the conjecture that AS coincides with pϵSOp. This would lead to a complete classification of QS, and is related to a conjecture about k-graphs.