9.29 Suppose there are 800 points on a circle, labeled $1, 2, \cdots$, 800 in a clockwise direction, dividing the circle into 800 arcs. Now, choose any one point and color it red, then proceed to color other points red according to the following rule: if the $k$-th point has been colored red, then move $k$ arcs in a clockwise direction and color the point reached red, and continue this process. How many red points can be obtained at most on the circle? Explain your reasoning.
[Solution] If we first paint point 1 red, we can obtain 25 red points $\{1,2,4,8,16, 32,64,128,256,512,224,448,96,192,384,768,736,672,544,288, 576,352,704,608,416\}$. This indicates that the maximum number of red points is no less than 25.

When the first red point is the $m$-th point, the set of all red points is $\left\{m 2^{k}(\bmod 800) \mid k=\right.$ $0,1,2, \cdots\}$. Since $2^{4}=16 \equiv L(\bmod 5)$, we have $2^{4 h} \equiv 1(\bmod 5)$. Therefore,
$$
2^{4} \equiv 16,2^{8} \equiv 6,2^{12} \equiv 21,2^{16} \equiv 11,2^{20} \equiv 1(\bmod 25)
$$

Thus, we get
$$
2^{25} \equiv 32 \quad(\bmod 800) .
$$

Therefore,
$$
\begin{array}{l}
\left\{m 2^{k}(\bmod 800) \mid k=0,1,2, \cdots\right\} \subset \\
\left\{m 2^{k}(\bmod 800) \mid k=0,1,2, \cdots, 24\right\} .
\end{array}
$$

This means that when we first paint the $m$-th point red, the number of all red points obtained does not exceed 25. In summary, the maximum number of red points is 25.