A function $f : \Bbb Z \times \Bbb Z \rightarrow \Bbb Z$ is defined as $f(u,v) = 3u + 6v.$ Is the function surjective? Prove it.

I had the following proof.

Proof
Pick $x = 2$, then $3u + 6v = 2 \Rightarrow 3(u + 2v) = 2$
Let $y = u + 2v$ $\exists y \in \Bbb Z \times \Bbb Z$.
Thus $3y = 2 \Rightarrow y = \frac{2}{3}$.
This is a contradiction because $\frac{2}{3} \not\in \Bbb Z \times \Bbb Z.$
The function is therefore not surjective.

I am a novice at this whole LaTex thing and relatively new to proofs and these surjective proofs are killing me I can't seem to get anything right. Any help is greatly appreciated.