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Lynna [10]
2 years ago
7

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

Mathematics
1 answer:
Grace [21]2 years ago
7 0
\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
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