NSEJS 2016

Number Theory Level pending

Find the number of integers $n$ such that $1+ n$ is a divisor of $1 + n^2$ .

The answer is 4.

1 solution

Brian Charlesworth
Apr 1, 2017

We require that $\dfrac{n^{2} + 1}{n + 1} = n - 1 + \dfrac{2}{n + 1}$ be an integer. This will be the case when $(n + 1)|2$ , and since $2$ has four divisors, namely $\pm1, \pm2$ , there will be $\boxed{4}$ such $n$ , namely $-3,-2, 0$ and $1$ .

suppose we apply remainder theorm : so, n+1 will divide n^2+1 only when n^2+1=0 when we put n = -1 but it equates to 2 so, we conclude that n+1 doesn't divide n^2+1

akarsh jain - 4 years, 2 months ago

NSEJS 2016

The answer is 4.

1 solution

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