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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