9.35 Find the largest positive integer $n$, such that there exists a unique integer $k$ satisfying
$$\frac{8}{15}<\frac{n}{n+k}<\frac{7}{13}$$
［Solution］From the assumption, we have $\frac{15}{8}>\frac{n+k}{n}>\frac{13}{7}$, thus $48 n<56 k<49 n$.
Therefore, the problem is reduced to finding the largest open interval $(48 n, 49 n)$, such that this interval contains only one multiple of 56. Since the length of such an interval is $n$, it contains $n-1$ integers. If $n-1 \geqslant 2 \times 56=112$, then the interval contains at least two multiples of 56. Therefore, we have $n \leqslant 112$.

On the other hand, when $n=112$, since
$$48 \times 112=96 \times 56<97 \times 56<98 \times 56=49 \times 112$$

there exists a unique $k=97$ that satisfies the given inequality.
Therefore, $n=112$.