Our goal is to count the number of subgroups of the direct product of finite cyclic groups, Zm _ Zn. When m and n are relatively prime, Zm _ Zn is isomorphic to Zmn. Counting the number of subgroups of Zmn is a simple exercise in combinatorics. However, if m and n are not relatively prime, then we use Goursat's (lesser-known) Theorem to find the number of subgroups of Zm _ Zn. We may also extend the usage of this theorem to cases beyond the direct product of finite cyclic groups.