$a,b,m,x$ are positive integers.
For which $x>0$ is $f(x)$ an integer?
$$f(x)=\frac{-b^2m-ba+ax}{-mx-bm-a}$$
I been trying to play with it, I changed it to:
$$\frac{b^2m-a\left(b+x\right)}{a+m\left(b+x\right)}$$
And then I been trying to say:
$$a+m\left(b+x\right)| b^2m-a\left(b+x\right) $$ $$a+m\left(b+x\right)| (a+m\left(b+x\right))(b+x)$$ So $$a+m\left(b+x\right)| b^2m-a\left(b+x\right)+ (a+m\left(b+x\right))(b+x)$$ $$a+m\left(b+x\right)| m\left(\left(b+x\right)^2+b^2\right)$$
But I don't see how it helps, so please help me.