15.14 $S$ is the set of rational numbers $r$ where $0<r<1$, and $r$ has a repeating decimal expansion of the form $\overline{0 . a b c a b c a b c \cdots}=\overline{0 . \dot{a} b \dot{c}}, a, b, c$ are not necessarily distinct. Among the elements of $S$, how many different numerators are there when written in lowest terms?
(10th American Invitational Mathematics Examination, 1992)
Given $999=3^{3} \cdot 37$.

If $\overline{a b c}$ is neither divisible by 3 nor by 37, then the fraction is in its simplest form. By the principle of inclusion-exclusion, we have
$$
999-\left(\frac{999}{3}+\frac{999}{37}\right)+\frac{999}{3 \cdot 37}=648
$$

such numbers.
In addition, there are fractions of the form $\frac{k}{37}$, where the natural number $k$ is a multiple of 3 and less than 37. Such $k$ values are
$3,6,9, \cdots, 36$, a total of 12.
Therefore, the number of numerators that satisfy the condition is $648+12=660$.