Find elements of a set that divide an expression.

Question

I have to determine the elements of the following set:

$A = \{x\in\ \mathbb Z \vert \sqrt[3]{\frac {7x + 2}{x+5}} \in \mathbb Z \}$

I know that $x+5 \not=0$ and $x+5$ must divide $7x + 2$ but I cannot figure out how to use the second condition and how to write down the complete solution step by step. Could someone, please, show me how it can be done? Thank you in advance.

Answer 1

What you said is correct. Use the fact that $7x+2 = 7(x+5) - 33$, and since $x+5$ divides $7x+2$, then $x+5$ must divide $33$, so you have

$x + 5 = \pm1, \pm3, \pm11, \pm33$. What is left is to check which of these are exact cubes.