Proof about set of representatives

Question

Let $X = \left\{(m, n)\in \mathbb{Z}\times\mathbb{Z}, n \neq 0\right\}$. Define a relation $\sim$ on $X$ by $(k, l) \sim (m, n)$ if $kn = lm$.

Prove that $\left\{(m, n)|m \in \mathbb{Z}, n \in \mathbb{N}, \text{gcd}(m, n) = 1\right\}$ is a complete set of representatives for the equivalence classes of $\sim$.

I don't know how to approach this problem.

I am trying to analyze specific pairs with $\text{gcd}(m,n)=1$. For example $(3,5)$ and $(3,4)$.

Let $(a,b)\sim (3,5)$, so $5a=3b$. I need to show that $(m,n)$ will be a unique pair from every class.

Answer 1

Let $(a,b)\in \mathbb{Z}\times \mathbb{Z}$ with $b\neq 0$. Let $m=\frac{\text{lcm}(a,b)}{b}$ and $n=\frac{\text{lcm}(a,b)}{a}$, then $mb=na$. Can you finish the argument?