Q: Order of an element in an external direct product Consider 
$\mathbb{Z}_{4}\times \mathbb{Z}_{4}=\left \{ 0,1,2,3 \right \}\times \left \{ 0,1,2,3 \right \}$
The element $\left ( 2,0 \right )$ is of order 2 but I cannot figure out why.
$2=LCM\left ( \left | 2 \right |,\left | 0 \right | \right )=LCM\left ( 2.2Mod4,0Mod4 \right )=LCM\left ( 2,0 \right )$
But $LCM\left ( 2,0 \right ) is....0$
Some clarification please? Thanks in advance

A: $(2,0)+(2,0)=(0,0)$ which is the identity.
Thus it's order is 2.
Another example: the order of $(1,1)$ is 4.
