Is ${\mathbb Z} \times {\mathbb Z}$ cyclic?

Question

Not sure where to go with this, but I don't think it is cyclic..

Answer 1

Hint: supose $\;m,n\in\Bbb Z\;$ are such that $\;\Bbb Z\times\Bbb Z=\langle (m,n)\rangle\;$, then among other things there must exist $\;x\in\Bbb Z\;$ s.t.

$$x(m,n)=(1,1)\implies xm=1=xn\implies\ldots ?$$

Well, now it must also be true that there exists $\;y\in\Bbb Z\;$ s.t. :

$$y(m,n)=(0,1)\implies ym=0\;,\;yn=1$$

So...

Answer 2

Suppose it were cyclic, with generator $(a,b)$. Then every element of ${\mathbb Z} \times {\mathbb Z}$ would have to be of the form $(na,nb)$ for some $n \in {\mathbb Z}$. Can that occur?