Given two integer variables $x$ and $y$. We are given that each integer variable $x$ and $y$ can't be greater than a given integer $z$.
The problem: We are given the proportions $a$ and $b$ such that $a + b = 1$, $a = \frac x z$, and $b = \frac y z$. Is it possible to reverse solve $x$ and $y$?
On
Note: This isn't a rigorous answer, and is only meant to get you on the right track. Work out the 'Why?' parts I've mentioned below.
From the information you have provided, since $a + b = 1$, $x+y=z$ for a given $z$, along with the constraints $x \leq z$ and $y \leq z$. Also, note that if $x=0, y=z$ and that $x=z$, when $y=0$. Therefore, $x,y \in [0,z]$ (Why?). It's easy to see that there would be $(z+1)$ solutions counting all such $x,y \in \mathbb Z.$(Why?)
Assuming you know $a$ and $z$:
From $a+b=1$, you can conclude ….
From $a=\frac x z$ you can conclude ….
From $b=\frac y z$ you can conclude ….