<s>[INST] Possible values of $N$ Find the number of values of $N$ such that the below expression is an integer:
 $(n+1)^2\over n+7$ is an integer
 [/INST] $(n+1)^2=n^2+2n+1=(n+7)(n-5)+36$
So, $\frac{(n+1)^2}{n+7}=n-5+\frac{36}{n+7}$
Assuming $n$ to be an integer, $(n+7)\mid36 \iff (n+7)\mid(n+1)^2$ 
So, $n+7$ can be any divisor  of $36,$ namely $\pm1,\pm2,\pm3,\pm4,\pm6,\pm9,\pm12,\pm18,\pm36$
If we constrain $n$ to be non-negative i.e., if $n+7\ge 7,$ then $n+7$ can be  $9,12,18,36$ 
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