5. (40 points) Find the largest positive integer $n$, such that there exists a unique positive integer $k$ satisfying $\frac{8}{15}<\frac{n}{n+k}<\frac{7}{13}$.
5. The inequality can be written as $1+\frac{7}{8}>1+\frac{k}{n}>1+\frac{6}{7}$. Or $\frac{98}{112}>\frac{k}{n}>\frac{96}{112}$.
If $n=112$, then the unique value of $k$ is 97.
Assuming $n>112$, then $\frac{98 n}{112 n}>\frac{112 k}{112 n}>\frac{96 n}{112 n}$.
Between $96 n$ and $98 n$, there are at least two numbers that are multiples of 112, in which case, the value of $k$ is not unique. Therefore, the maximum value of $n$ is 112.