38 There are 800 points on a circle, labeled $1,2, \cdots, 800$ in a clockwise direction, dividing the circle into 800 gaps. Now, choose one of these points and color it red, then perform the following operation: if the $k$-th point is colored red, then move $k$ gaps in a clockwise direction and color the point reached red. Question: According to this rule, what is the maximum number of points that can be colored red on the circle? Prove your conclusion.
38. This is equivalent to finding the maximum number of distinct numbers in the sequence $a, 2a, 2^2a, 2^3a, \cdots$ under modulo 800, where $a$ takes values in $1, 2, \cdots, 800$.

Notice that when $2^n \not\equiv 2^m \pmod{800}$, it is not necessarily true that $2^n a \not\equiv 2^m a \pmod{800}$. Conversely, when $2^n a \not\equiv 2^m a \pmod{800}$ holds, $2^n \equiv 2^m \pmod{800}$ must hold. Therefore, the maximum number of distinct elements in the sequence $a, 2a, 2^2a, \cdots$ under modulo 800 can be achieved when $a=1$. Thus, we only need to find the number of distinct elements in the sequence $1, 2, 2^2, \cdots$ under modulo 800.

Since $800 = 2^5 \times 5^2$, and for $n \geq 5$ we have
$$2^n \equiv 0 \pmod{2^5}$$

Additionally, $\{2^n \pmod{25}\}$ is: $2, 4, 8, 16, 7, 14, 3, 6, 12, -1, -2, -4, -8, -16, -7, -14, -3, -6, -12, 1, \cdots$, so $\{2^n \pmod{25}\}$ contains exactly 20 distinct elements. Combining this with $\{2^n \pmod{2^5}\}$ being $2, 4, 8, 16, 0, 0, \cdots$, we get that $\{2^n \pmod{800}\}$ contains $20 + 4 = 24$ (distinct) numbers.

Therefore, there are at most 24 points on the circle that are colored red.