Hw9 Problem 2

What is the largest order among the orders of all cyclic subgroups of $\mathbb{Z_{15}} \times \mathbb{Z_{25}}$.
By Theorem 11.5, $\mathbb{Z_{15}} \times \mathbb{Z_{25}}$ is not cyclic because $15$ and $25$ are not relatively prime and by Definition 11.8, the largest order among the orders of all cyclic subgroups is equal to $lcm(15,25) = 75$.