17. (18 points) Among 200 small balls numbered $1, 2, \cdots, 200$, any $k$ balls are drawn such that there must be two balls with numbers $m$ and $n$ satisfying
$$
\frac{2}{5} \leqslant \frac{n}{m} \leqslant \frac{5}{2} \text {. }
$$

Determine the minimum value of $k$ and explain the reasoning.
17. Divide 200 balls numbered $1 \sim 200$ into 6 groups, such that the ratio of the numbers of any two balls in each group is no less than $\frac{2}{5}$ and no more than $\frac{5}{2}$.
Grouping as follows:
Group 1 $(1,2)$
Group 2 $\quad(3,4,5,6,7)$
Group 3 $(8,9,10, \cdots, 20)$
Group 4 $\quad(21,22,23, \cdots, 52)$
Group 5 $\quad(53,54,55, \cdots, 132)$
Group 6 $\quad(133,134,135, \cdots, 200)$
When $k=7$, among the 200 balls in these 6 groups, if 7 balls are randomly drawn, according to the pigeonhole principle, there must be two balls in the same group, and the numbers of these two balls are $m$ and $n$, which must satisfy
$$
\frac{2}{5} \leqslant \frac{n}{m} \leqslant \frac{5}{2} .
$$

Therefore, the minimum value of $k$ is 7.