19.18 The ratio of the interior angles of two regular polygons is $3: 2$. The number of such regular polygons is
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(E) infinitely many.
(13th American High School Mathematics Examination, 1962)
[Solution] Let $N, n$ represent the number of sides of the two regular polygons, respectively. Therefore, the relationship between their angle measurements can be expressed as
$$
\frac{(N-2) 180^{\circ}}{N}=\frac{3}{2} \cdot \frac{(n-2) 180^{\circ}}{n},
$$

then $N=\frac{4 n}{6-n}$.
Since $n, N$ are both positive integers no less than 3, we only need to consider the cases where $n=3,4,5$, which correspondingly yield $N=4,8,20$.
(Note that when $n=6$, $6-n=0$; when $n>6$, $6-n<0$) Therefore, the correct choice is $(C)$.