Q. Show that $\mathbb{Z\_3 x Z\_4}$ is a cyclic group.

So my question is there a faster way besides listing all the elements and besides knowing the theorem. 

Since the process I am doing is:

I know:
$\mathbb{Z\_3 x Z\_4}=\{(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3),(2,0),(2,1),(2,2),(2,3)\}$

and I've been taking each element and figuring out the order such as:

$$(1,0): (1,0); (1,0)+(1,0)=(2,0);(2,0)+(1,0)=(0,0)$$ which is order 3.

and I believe I have to keep doing this until I find the element that gives me order 12.

but I realize that is kinda tedious. So my question is, is there a simpler way or would be it suggested to continue on the method I am doing.