show a quotient group is cyclic

Question

Let a and b be nonzero integers. Let $\sigma:\mathbb{Z}\times \mathbb{Z}\to\mathbb{Z}$ and $\sigma((x,y))=ax+by.$ Suppose $\gcd(a,b)=1$. Show that $(\mathbb{Z}\times \mathbb{Z})/\langle(a,b)\rangle$ is cyclic.

It would be great if anyone can let me know what I should do to prove it.

Thanks

Answer 1

Or I'm misunderstanding something, or something's missing in the question...or the question is completely trivial: we have that

$$\Bbb Z\times\Bbb Z/\langle (a,b)\rangle \cong H\le \Bbb Z$$

But $\;\Bbb Z\;$ is cyclic and thus also $\;H\le \Bbb Z\;$ is cyclic...

Added on request: Since g.c.d.$\,(a,b)=1\;$ then there exist $\;m,n\in\Bbb Z\;$ s.t. $\;ma+nb=1\;$ , and thus

$$\sigma (m,n)=ma+nb=1\implies \text{Im}\,\sigma=\Bbb Z$$