6. The largest positive integer $n$ for which the inequality $\frac{8}{15}<\frac{n}{n+k}<\frac{7}{13}$ holds for a unique integer $k$ is $\qquad$ .
6.112 .

Obviously, $\frac{13}{7}<\frac{n+k}{n}<\frac{15}{8} \Leftrightarrow \frac{6}{7}<\frac{k}{n}<\frac{7}{8}$
$$
\Leftrightarrow 48 n<56 k<49 n \text {. }
$$
There are at least $n-1$ integers between $48 n+1$ and $49 n-1$.
If $n-1 \geqslant 2 \times 56$, then there must be at least 2 multiples of 56 in this range, so, $n-1<2 \times 56$, which means $n \leqslant 112$.
Also, $48 \times 112=56 \times 96<56 \times 98=112 \times 49$, hence $k=97, n=112$.