<s>[INST] For what integer $n$ is $1+n$ divisible by $1+n^2$? For what integer $n$ is $1+n$ divisible by $1+n^2$?
I tried it in modulo $3$ and $8$ but to no avail. 
Trial and error gave me answers $0$,$1$,$-2$ and $-3$.
 [/INST] HINT:
$$\dfrac{n^2+1}{n+1}=\dfrac{(n+1-1)^2+1}{n+1}=n+1-2+\dfrac2{n+1}$$
OR
$$\dfrac{n^2+1}{n+1}=\dfrac{(n+1)(n-1)+2}{n+1}=n-1+\dfrac2{n+1}$$
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