2022 AIME I Problems/Problem 13
Contents
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 that (with one of more
such that
) can meet the requirement
.
Make cases by factors of . (A venn diagram of cases would be nice here.)
Case :
and
and
, aka
.
Euler's totient function counts these:
values (but it's enough to note that it's a multiple of 1000 and thus does not contribute to the final answer).
The remaining cases have (or
),
, and/or
as factors of
, which cancel out part of
.
Note: Take care about when to use
vs
!
Case :
, but
and
.
Then to leave 3 uncancelled, and
,
so
, giving:
,
,
,
for a subtotal of values.
Case :
, but
and
.
Much like previous case, is
, so
,
giving values.
Case :
and
(so
), but
.
Here, is
, so
,
giving values.
Case :
.
Here, is
, so
,
giving values, so we don't need to account for multiples of
and
.
To sum up, the answer is
Video Solution
https://MathProblemSolvingSkills.com
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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