2018 AMC 12B Problems/Problem 17
Problem
Let and
be positive integers such that
and
isi as small as possible. What is
?
Solution 1
We claim that, between any two fractions and
, if
, the fraction with smallest denominator between them is
. To prove this, we see that
which reduces to
. We can easily find that
, giving an answer of
. (pieater314159)
Solution 2 (requires justification)
Assume that the difference results in a fraction of the form
. Then,
Also assume that the difference
results in a fraction of the form
. Then,
Solving the system of equations yields and
. Therefore, the answer is
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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All AMC 12 Problems and Solutions |
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