$16 \cdot 9$ If 20 points divide a circle into 20 equal parts, the number of regular polygons that can be formed with vertices chosen only from these 20 points is
(A) 4 .
(B) 8 .
(C) 12 .
(D) 24 .
(China Junior High School Mathematics League, 1996)
[Solution] Let a regular $k$-sided polygon satisfy the condition, then the $20-k$ points other than the $k$ vertices are uniformly distributed on the minor arcs opposite to the sides of the regular $k$-sided polygon.
Thus, $\frac{20-k}{k}=\frac{20}{k}-1$ is an integer, so $k \mid 20$.
But $k \geqslant 3$,
$\therefore k=4$ or 5 or 10 or 20.
$\therefore$ The number of regular polygons sought is:
$$
\frac{20}{4}+\frac{20}{5}+\frac{20}{10}+\frac{20}{20}=12 \text { (polygons). }
$$

Therefore, the answer is $(C)$.