Question

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

Answer 1

`(n^2+n+1)/(n-1)=((n-1)^2+3(n-1)+3)/(n-1)=n-2+\frac3{n-1}`

The remainder term
`\frac3{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`