Solving for a variable in an integer divisibility problem Say I have a problem of the form
$$\frac a{bx+c}=z$$
Where $a,b,c$ are known integers, $x$ is some unknown variable , and $z$ is an integer output.. Is there an approach I could take to determine if there is some integer $x$
I was thinking of something along the lines of modular arithmetic of the form
$a = 0 \pmod{bx+c}$
But any effort to determine  leads me in circles. Am I thinking along the right lines? Is what I am attempting to do possible without resorting to trial and error?

Suppose that $b\not=0$.
$bx+c$ has to be a divisor of $a$. So, letting $d$ be a divisor of $a$, we have
$$bx+c=d\iff bx=d-c\iff x=\frac{d-c}{b}.$$
Hence, $x$ is an integer if and only if $b$ is a divisor of $d-c$. 
As a result, there exists such an integer $x$ if and only if there exists a divisor $d$ of $a$ such that 
$$\text{$b$ is a divisor of $d-c$.}$$
