14. [9] Compute the sum of all positive integers $n$ for which
$$
9 \sqrt{n}+4 \sqrt{n+2}-3 \sqrt{n+16}
$$
is an integer.
Answer: 18
For the expression to be an integer at least one of $n$ and $n+2$ must be a perfect square. We also note that at most one of $n$ and $n+2$ can be a square, so exactly one of them is a square.
Case 1: $n$ is a perfect square. By our previous observation, it must be that $4 \sqrt{n+2}=3 \sqrt{n+16} \Rightarrow$ $n=16$.
Case 2: $n+2$ is a perfect square. By our previous observation, it must be that $9 \sqrt{n}=3 \sqrt{n+16} \Rightarrow$ $n=2$.
Consequently, the answer is $16+2=18$.