Example 1 Find the smallest positive integer $n$ such that for this $n$, there is a unique positive integer $k$ satisfying
$$
\frac{8}{15}<\frac{n}{n+k}<\frac{7}{13} \text {. }
$$
(Fifth American Mathematical Invitational)
Solution: Since the fractions $\frac{8}{15}$ and $\frac{7}{13}$ satisfy $15 \times 7 - 8 \times 13 = 1$, it must be that $\frac{n}{n+k} = \frac{8+7}{15+13} = \frac{15}{28}$, thus we get $n=15, k=13$.