# Problem #220

 220 Let $S^{}_{}$ be the set of all rational numbers $r^{}_{}$, $0^{}_{}, that have a repeating decimal expansion in the form $0.abcabcabc\ldots=0.\overline{abc}$, where the digits $a^{}_{}$, $b^{}_{}$, and $c^{}_{}$ are not necessarily distinct. To write the elements of $S^{}_{}$ as fractions in lowest terms, how many different numerators are required? This problem is copyrighted by the American Mathematics Competitions.
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