# 1992 AIME Problems/Problem 5

## Problem

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?

## Solution

We consider the method in which repeating decimals are normally converted to fractions with an example: $x=0.\overline{176}$ $\Rightarrow 1000x=176.\overline{176}$ $\Rightarrow 999x=1000x-x=176$ $\Rightarrow x=\frac{176}{999}$

Thus, let $x=0.\overline{abc}$ $\Rightarrow 1000x=abc.\overline{abc}$ $\Rightarrow 999x=1000x-x=abc$ $\Rightarrow x=\frac{abc}{999}$

If $abc$ is not divisible by $3$ or $37$, then this is in lowest terms. Let us consider the other multiples: $333$ multiples of $3$, $27$ of $37$, and $9$ of $3$ and $37$, so $999-333-27+9 = 648$, which is the amount that are neither. The $12$ numbers that are multiples of $81$ reduce to multiples of $3$. There aren't any numbers which are multiples of $372$, so we can't get numerators which are multiples of $37$. Therefore $648 + 12 = \boxed{660}$.

 1992 AIME (Problems • Answer Key • Resources) Preceded byProblem 4 Followed byProblem 6 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions