A and B are groups with binary operations $\displaystyle *$ and ?

Prove that the order of $\displaystyle a\times b$ is $\displaystyle \text{LCM}[|a|,|b|]$

The binary operation of $\displaystyle A\times B$ is define as such:

$\displaystyle (a\times b)(c\times d)=(a*c)\times (b\text{?} d)$

Let $\displaystyle |a|=n$ and $\displaystyle |b|=m$.

Do I need to define the gcd as well? Regardless if I do or not, I am not sure where to proceed next.