Proving that something is not cyclic

Question

Let $m,n \in \Bbb N$ prove that if $gcd(m,n)>1$ then $\Bbb Z_m\times\Bbb Z_n$ is not cyclic.

My solution attempt :I know that if $gcd(m,n)=1$ then $\Bbb Z_m\times\Bbb Z_n$ is cyclic so how would i use this to prove that if it is $>1$ then it is not cyclic?

Answer 1

If $\gcd(m,n) >1$ then $L:=\text{lcm}(m,n) = \dfrac{mn}{\gcd(m,n)} < mn$.

Since $m\mid L$ and $n\mid L$ then $L(x,y) = (Lx,Ly) = (0,0)$ for all $(x,y)\in \mathbb{Z}_m\times\mathbb{Z}_n$. Therefore there's no element of degree $mn$, i.e. $\mathbb{Z}_m\times\mathbb{Z}_n$ is not cyclic.

Answer 2

Hint: Let $l=lcm(m,n)$. Then $la=0$ for all $a \in \Bbb Z_m\times\Bbb Z_n$.