Question

Find all integers $n$ for which $\frac{n^2+n+1}{n-1}$ is an integer.

Answer 1

$\dfrac{n^2+n+1}{n-1}=\dfrac{(n-1)^2+3(n-1)+3}{n-1}=n-2+\dfrac3{n-1}$

The remainder term $\dfrac3{n-1}$ will never vanish, and will always be rational as long as the denominator is not 1, 3, or -3.

$n-1=1\implies n=2$
$n-1=3\implies n=4$
$n-1=-3\implies n=-2$