1. Let $S$ be the set of all integers $n$ such that $\frac{8 n^{3}-96 n^{2}+360 n-400}{2 n-7}$ is an integer. Find the value of $\sum_{n \in S}|n|$.
1. Answer: 50
Note that $\frac{8 n^{3}-96 n^{2}+360 n-400}{2 n-7}=4 n^{2}-34 n+61+\frac{27}{2 n-7}$. Since $4 n^{2}-34 n+61$ is an integer for all integers $n$, we must have that 27 divisible by $2 n-7$. Hence, $2 n-7=$ $-1,1,-3,3,-9,9,-27,27$, so that $n=3,4,2,5,-1,8,-10,17$. Hence the required sum equals 50 .