If $a,b,x,y\in\Bbb N$ , and $ax-by=(a,b)$, then $(x,y)=1$

Question

Need to prove an exercise, and for that I need to show that if $$a,b,x,y\in\Bbb N$$ and
$$ax-by=(a,b),$$then$$(x,y)=1.$$

How to do this? I have no idea. Please do not use modular arithmetic.

Answer 1

If $ax - by = (a, b)$ then $$\dfrac{a}{(a, b)}x - \dfrac{b}{(a, b)}y = 1.$$ Dividing by $(x, y)$ we see that $$\frac{a}{(a, b)}\frac{x}{(x, y)} - \frac{b}{(a, b)}\frac{y}{(x, y)} = \frac{1}{(x, y)}.$$ Now note that all of the fractions on the left hand side are actually integers, so the right hand side must be an integer. In particular, $(x, y) = \pm 1$, but $(x, y) \geq 1$ so $(x, y) = 1$.