Relation $ab=cd$ in $\mathbb{Z}$(UFD) with $a$ and $c$ coprime.

Question

Why if $15x=-19y$ where $x$ and $y$ are integers, this means that there is an integer t such that x=-19t and y=15t. I think it has something to do with the fact that $\mathbb{Z}$ is a UFD, but I can't understand why.

Answer 1

You have Gauß'lemma (generalisation of Euclid's lemma):

If an integer $a$ divides $bc$ ($b,c\in\mathbf Z$) ans if $a$ and $b$ are coprime, then $a$ divides $c$.

Naturally, this is valid in any U.F.D.

Answer 2

You have $$15x =- 19y$$ where $x$ and $y$ are integers.

Well since $15|15x$ it has to divide $-19y$ and since $15$ and $-19$ are relatively prime, $15$ has to divide $y$.

That is $y=15t$

Now if you plug $y=15t$ in your original equation and divide both sides by $15$ you will find that $x=-19t$

Answer 3

Yes, discarding the trivial case $\,x =0\iff y=0\,$ we can write it fractionally as

$$\dfrac{x}y = \dfrac{-19\ \ }{15}\iff \begin{align} &x = -9n\\ &y =\ 15n\end{align}\ \ \text{ for some integer } n$$

This property - well known since grade school - remains true if we replace $-19/15\,$ by any other irreducible fraction. A proof follows easily from existence and uniqueness or prime factorizations (UFD) or closely related results such as Euclid's Lemma, Four Number Theorem, etc. It is sometimes called unique fractionization. See this answer for further discussion.