2022 AIME I Problems/Problem 13
Problem
Let be the set of all rational numbers that can be expressed as a repeating decimal in the form where at least one of the digits or is nonzero. Let be the number of distinct numerators obtained when numbers in are written as fractions in lowest terms. For example, both and are counted among the distinct numerators for numbers in because and Find the remainder when is divided by
Solution
, .
Then we need to find the number of positive integers less than that can meet the requirement. Suppose the number is .
Case : . Clearly satisfies.
Case : but is not a multiple of or Then the least value of is , so that , values from to
Case : but is not a multiple of or . Then the least value of is , so that , values from to .
Case : . None.
Case : . Then the least value of is , values from to
To sum up, the answer is mod .
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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