Prove that the characteristic of Z/mZ × Z/nZ equals D, where D is the least common multiple between m and n

Question

I have no idea to go about the proof. Intuitively, I understand that the characteristic is the point where elements start repeating themselves, and of course for the elements in Z/m X Z/n to start repeating themselves, it will only occur once we hit their least common multiple.

I just don't know how to write out the proof formally or what the notation would be.

Answer 1

Hint: Let $D$ denote the least common multiple of $m$ and $n$.

• Show that the element $(1,1)$ has order $D$.
• Show that for any $g \in \Bbb Z_m \times \Bbb Z_n$, $g^D = (0,0)$.