$\displaystyle A = \{0, 1, 2, 3, 4\}\and\B = \{0, 1, 2, 3\},a\epsilon A \and\ b \epsilon B$

I need to find the Relation such that $\displaystyle R = \{(a, b)|\gcd(a, b)=1\}$

I included (0, 0) as one of my ordered pairs in the relation set, but apparently it isn't suppose to be in it. Why is that? Isn't 0 divisible by one?

And for the same two sets, A and B, I have to find the relation $\displaystyle R=\{(a,b)|lcm(a, b)=2\}$

What I found was $\displaystyle R=\{(2,2),(4,2)\}$

But the only correct ordered-pair in that set is (2, 2). I'm not entirely sure what I did wrong...