1992 AIME Problems/Problem 5
Problem
Let be the set of all rational numbers , , that have a repeating decimal expansion in the form , where the digits , , and are not necessarily distinct. To write the elements of 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:
Thus, let
If is not divisible by or , then this is in lowest terms. Let us consider the other multiples: multiples of , of , and of both and , so , which is the amount that are neither. The numbers that are multiples of reduce to multiples of . We have to count these since it will reduce to a multiple of which we have removed from , but, this cannot be removed since the numerator cannot cancel the .There aren't any numbers which are multiples of , so we can't get numerators which are multiples of . Therefore .
1992 AIME (Problems • Answer Key • Resources) | ||
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