Darij Grinberg answered the question in a comment, but I thought a more geometric interpretation might be helpful.

The group of positive rational numbers under multiplication is isomorphic to an infinite sum of copies of $\mathbb{Z}$, where the set of primes forms a natural basis. If you choose a positive integer $N$, you can think of the subgroup $N^\mathbb{Z}$ as a copy of the integers embedded as a line in an infinite dimensional lattice, with the generator $N$ given in coordinates by its prime factorization. Torsion elements of the quotient group $\mathbb{Q}_{>0}^\times/N^\mathbb{Z}$ must then lift to elements in the lattice that "lie between" elements of $N^\mathbf{Z}$. The torsion part of the quotient group is then generated by $N^{1/g}$, where $g$ is the largest integer such that $N$ is a perfect $g$th power.