1. Compute the sum of all positive integers $n$ for which the expression
$$
\frac{n+7}{\sqrt{n-1}}
$$
is an integer.
Answer: $\square$
$\square$
Solution: We know $\sqrt{n-1}$ must be a positive integer, because the numerator is a positive integer, and the square root of an integer cannot be a non-integer rational. From this,
$$
\frac{n+7}{\sqrt{n-1}}=\sqrt{n-1}+\frac{8}{\sqrt{n-1}}
$$
is a positive integer, so we $\sqrt{n-1}$ must be a positive integer that divides 8 . There are 4 such positive integers: $1,2,4,8$, which give $n=2,5,17,65$, so the answer is 89 .