On a basic tensor product question I am trying to show that
$$
\mathbb{Z} / (10) \otimes \mathbb{Z} / (12) \cong \mathbb{Z}/(2)
$$
by defining a map
$$
h([a]_{10} \otimes [b]_{12}) = [ab]_2
$$
and extend it linearly. I am having trouble trying to prove that this map is 
well defined on $\mathbb{Z} / (10) \otimes \mathbb{Z} / (12)$. (I am using the 
constructive definition of tensor products for exercise). I would greatly appreciate any help!
Thanks!

try to show in general that: let   $A$ a commutative ring and $I$ , $J$ two ideals, then $A/I\otimes A/J\simeq A/(I+J)$. And remark that $12\mathbb{Z}+10\mathbb{Z}=2\mathbb{Z}$.   
