A number theory problem by A Former Brilliant Member

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

As $n - 1$ is an integer, we see that $(n + 1)|(n^{2} + 1)$ if and only if $(n + 1)|2$ , which for $n \gt 0$ is only the case when $n = 1$ , and so the number of positive integers that satisfy the given condition is $\boxed{1}$ .

We should be able to show that $n+1 \;|\; (n^2+1)-(n+1)(n-1)=2$ which says $n+1|2$ since $n$ is positive the only possibility is $\boxed{n=1}$

There is only one such positive integer: $n=1$ .