Question

Find the sum of all positive integers & such that
x³ - x + 120\(x - 1)(x+1)
is an integer.​

Answer 1

We have to find the solution for

x3−x+15(x−1)(x+1)∈Zx3−x+15(x−1)(x+1)∈Z where x∈Zx∈Z

∴x3−x+15x2−1∈Z∴x3−x+15x2−1∈Z where x∈Zx∈Z

∴x+15x2−1∈Z∴x+15x2−1∈Z where x∈Zx∈Z

We now need to find the positive integer values of xx such that 15x2−115x2−1 is an integer.

15x2−115x2−1 can be an integer only if x2−1x2−1 is a factor of 1515 .

So the possible values of x2−1x2−1 are ±1±1 , ±3±3 , ±5±5 and ±15±15

Solving the above equations, the positive integer values of xx are 22 and 44 .

And their sum is 66 .