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 1
,
.
Then we need to find the number of positive integers less than 10000 can meet the requirement.Suppose the number is x.
Case 1: (9999, x)=1. Clearly x satisfies.
Case 2: 3|x but x is not a multiple of 11 or 101. Then the least value of abcd is 9x, so that , 334 values from 3 to 1110.
Case 3: 11|x but x is not a multiple of 3 or 101. Then the least value of abcd is 11x, so that , 55 values from 11 to 902.
Case 4: 101|x. None.
Case 5: 3, 11|x. Then the least value of abcd is 11x, 3 values from 33 to 99.
To sum up, it is .
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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