Densities of arithmetic Hecke triangle group orbits

We give new proofs computing the asymptotic densities for orbits of the arithmetic Hecke triangle groups ${\mathrm{\Gamma }}_q$ when $q=4$ and $q=6$. We use elementary number theory techniques along with basic properties of the M"obius function and Riemann zeta function with additional congruence conditions. This note actually came about by observing that, in the $q=4$ case, the underlying congruence condition partitions the set of coprime integer pairs into three classes of equal density.