Here is the problem I'm working on, from an old qualifying exam:

Let M be the $\displaystyle \mathbb{Z}$-module generated by a, b, c with the relations $\displaystyle 4a+3b+3c=2a-b+3c=0$. Express M as a direct sum of cyclic modules. What are the orders of these modules?

There's only two relations, so I'm guessing that one of the modules is going to be $\displaystyle \mathbb{Z}$. I managed to find a vector that is perpendicular to (4,3,3) and (2,-1,3), which is (6, -3, -5). I'm not sure what to do with it though, or if I'm even going in the right direction.

Thanks,

Hollywood