Good integers

Number Theory Level 3

Integer $n$ is such that $\dfrac{n^2 + 1}{n + 1}$ is also an integer.

If the sum of all such values of $n$ is $m$ , find $|m|$ .

1 solution

Mr. India
Feb 19, 2019

$\frac{n^2 + 1} {n + 1}$ is integer which means

$\frac{n^2 - 1 + 2}{n + 1}$ is integer

Or $\frac{(n + 1)(n - 1)}{n + 1}$ + $\frac{2}{n + 1}$ is integer.

Therefore n - 1 + $\frac{2}{n + 1}$ is integer.

As n - 1 is already integer, $\frac{2}{n + 1}$ needs to be integer.

Factors of 2 : 1, 2, - 1, - 2

n + 1 = 1, 2, - 1, - 2

Or n = 0, 1, - 2, - 3

Therefore m = - 4

Or $\boxed{ |m| = 4}$