how to prove that $ax + b$ is not divisible by $cx$

Question

How to prove that there exist no positive integer $x$ that satisfy the fraction

$$\frac{ax + b}{cx} = n$$

with $n$ a positive integer

Example : $a = 7$ ; $b = 558$ ; $c = 23$

Answer 1

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If $ax + b \over cx $ $=n$

While $n$ must be a positive integer

Then the top part of the fraction, $ax+b$ must be a multiple of $x$ for n to be a positive integer, since the bottom part, $cx$ is also a multiple of $x$.

If $x$ is 5, then $b$ must be 5, because a multiple of x ($ax$) plus $x$ (in this case, $b$) must be another multiple of $x$.

For this equation to work, $x = b$ while $c = a+1$.

Any three numbers can work with this equation. However, you didn't specify whether you can have two unknowns representing the same number. If your question does not allow two unknows representing the same number, then no positive integer can fulfil this equation, since $x$ has to be equal to $b$.