For which positive integers $n$ does the number $n^{2}+1$ divide $n+1$?
The integers considered are strictly positive. Thus, if $\mathfrak{n}^{2}+1$ divides $n+1$, then $n^{2}+1 \leqslant n+1$, hence $n^{2} \leqslant n$. And since $n>0, n \leqslant 1$. Finally, $n$ being an integer, the only candidate left is $n=1$, and we verify that it works: 2 divides 2. It's good!

The following exercises aim to show the importance of factorization in exercises involving divisibility.