When $f(x) = \frac{ax + b}{a -x + 1}$ is an integer

Question

Given that $a$ and $b$ are positive integers. and $$f(x) = \frac{ax + b}{a -x + 1}$$ is an integer, what integer values can x have?

If I could only somehow move $x$ from numerator to the denominator I would be able to solve this by factoring the numerator.

Answer 1

Since $$a-x+1\mid ax+b \;\;\;{\rm and} \;\;\;a-x+1\mid a(a-x+1)$$ we have $$a-x+1\mid (ax+b)+(a^2-ax+a)= a^2+a+b$$ So $x=a-d+1$ where $d$ divides $a^2+a+b$.