We investigate the K-theory of unital UCT Kirchberg algebras Q S arising from families S of relatively prime numbers. It is shown that K * (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 A S of Q S . For the K-theory of 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 ∈ S} plays a central role as well. In the case where |S| ≤ 2 or g S = 1 we obtain a complete classification for Q S . Our results support the conjecture that A S coincides with ⊗ p∈S Op. This would lead to a complete classification of Q S , and is related to a conjecture about k-graphs.Datethen we get a complete classification for Q S with the rule that Q S and Q T are isomorphic if and only if |S| = |T | and g S = g T , see Conjecture 6.5.At a later stage, the authors learned that Li and Norling obtained interesting results for the multiplicative boundary quotient for N H + by using completely different methods, see [LN16, Subsection 6.5]. Briefly speaking, the multiplicative boundary quotient related to Q S is obtained by replacing the unitary u by an isometry v, see Subsection 2.2 for details. As a consequence, the K-theory of the multiplicative boundary quotient does not feature a non-trivial free part. It seems that A S is the key to reveal a deeper connection between the K-theoretical structure of these two C * -algebras. As this is beyond the scope of the present work, we only note that the inclusion map from A S into Q S factors through the multiplicative boundary quotient as an embedding of A S and the natural quotient map. The results of [LN16] together with our findings indicate that this embedding might be an isomorphism in K-theory. This idea is explored further in [Sta, Section 5].The paper is organized as follows: In Section 2, we set up the relevant notation and list some useful known results in Subsection 2.1. We then link Q S to boundary quotients of right LCM semigroups, see Subsection 2.2, and a-adic algebras, see Subsection 2.3. These parts explain the central motivation behind our interest in the K-theory of Q S . In addition, the connection to a-adic algebras allows us to apply a duality theorem from [KOQ14], see Theorem 3.1, making it possible to invoke real dynamics. This leads to a decomposition result for K * (Q S ) presented in Section 4, which essentially reduces the problem to determining the K-theory of A S . The structure of the torsion subalgebra A S is discussed in Section 5. Finally, the progress on the classification of Q S we obtain via a spectral sequence argument for the K-theory of A S is presented in Section 6.Remark 5.9. Similar to Λ S,θ , we can also consider the row-finite k-graph Λ S,σ with σ p,q being the flip, i.e. σ p,q (m, n) := (n, m). That is to say, we keep the skeleton of Λ S,θ , but replace θ by σ. In thi...
