Question

Find the sum of all positive integers $x$ such that $\frac{x^3-x+120}{(x-1)(x+1)}$ is an integer.
(correct answer + 2, wrong answer 0)

Answer 1

We have,
$\frac{x^3-x+120}{(x-1)(x+1)}$
$=x+\frac{120}{x^2-1}$
It is an integer if and only if $x^2-1$ divides $120$
Then $x^2-1=\pm 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120$
Hence, $x=0,2,3,4,5,11$ and $2+3+4+5+11=25$