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 | ||
All AIME Problems and Solutions |
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