Normal Subgroups and Quotient Groups · Hard problem

In $\mathbb Z_{90}$, let $K = \langle 9\rangle$. Of the two candidates $N = \langle 18\rangle$ and $N' = \langle 15\rangle$, exactly one satisfies $N \le K$, as the Third Isomorphism Theorem requires. For that valid subgroup, what is $|K/N|$?