37 points are arranged on a circle, and one of the points is labeled with the number 1. Moving clockwise, label the next point with the number 2 after counting two points, then label the next point with the number 3 after counting three points, and so on, labeling the points with the numbers $1,2, \cdots, 2000$. In this way, some points are not labeled, and some points are labeled with more than one number. Question: What is the smallest number among the numbers labeled on the point that is labeled with 2000?
37. Equivalent to finding the smallest positive integer $n$, such that
$$1+2+\cdots+n \equiv 1+2+\cdots+2000(\bmod 2000)$$

i.e. $\square$
$$\frac{n(n+1)}{2} \equiv 1000(\bmod 2000),$$

which is equivalent to
$$n(n+1) \equiv 2000(\bmod 4000),$$

This requires
$$2000 \mid n(n+1)$$

Notice that
$$(n, n+1)=1$$

and
$$2000=2^{4} \times 5^{3},$$

so $2^{4}\left|n, 5^{3}\right| n+1$; or $5^{3}\left|n, 2^{4}\right| n+1$; or one of $n$ and $n+1$ is a multiple of 2000. Solving these, the smallest $n$ is found to be $624, 1375, 1999$, with the smallest number satisfying (1) being 624. Therefore, the smallest number marked on the point labeled 2000 is 624.