2011 AMC 10B Problems/Problem 24
Contents
Problem
A lattice point in an -coordinate system is any point
where both
and
are integers. The graph of
passes through no lattice point with
for all
such that
. What is the maximum possible value of
?
Solution 1
We see that for the graph of to not pass through any lattice points, the denominator of
must be greater than
, or else it would be canceled by some
which would make
an integer. By using common denominators, we find that the order of the fractions from smallest to largest is
. We can see that when
,
would be an integer, so therefore any fraction greater than
would not work, as substituting our fraction
for
would produce an integer for
. So now we are left with only
and
. But since
and
, we can be absolutely certain that there isn't a number between
and
that can reduce to a fraction whose denominator is less than or equal to
. Since we are looking for the maximum value of
, we take the larger of
and
, which is
.
Solution 2
We want to find the smallest such that there will be an integral solution to
with
. We first test A, but since the denominator has a
,
must be a nonzero multiple of
, but it then will be greater than
. We then test B.
yields the solution
which satisfies
. We know that MAA orders the answers in ascending order, so therefore we know that the smallest possible
must be
See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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