1992 AIME Problems/Problem 5

Problem

Let $S^{}_{}$ be the set of all rational numbers $r^{}_{}$ , $0^{}_{}<r<1$ , 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 $3$ . We have to count these since it will reduce to a multiple of $3$ which we have removed from $999$ , but, this cannot be removed since the numerator cannot cancel the $3$ .There aren't any numbers which are multiples of $37^2$ , so we can't get numerators which are multiples of $37$ . Therefore $648 + 12 = \boxed{660}$ .

Solution 2

We can see that the denominator of the fraction will be 999b, where b is either 1, 1/3, 1/111, 1/9, or 1/333. This means that the numerator stays abc unless abc divides 999. Finding the numbers relatively prime to 999 gives us the numerators when the fraction cannot be simplified past 999.

$\Rightarrow 999(1 - \frac{1}{3})(1- \frac{1}{111}) = 660$

Now, look at a number that does divide 999:

$\Rightarrow 555/999 = 5/9$

We can see that if a number does divide 999, it will be simplified into a number which is relatively prime to 999 (since that is what simplifying is). There are 660 numbers less than or relatively prime to 999, so the answer is $\boxed{660}$ . - krishkhushi09

1992 AIME (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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1992 AIME Problems/Problem 5

Problem

Solution

Solution 2