Problem:
A lattice point in an xy-coordinate system is any point (x,y) where both x and y are integers. The graph of y=mx+2 passes through no lattice point with 0<x≤100 for all m such that 21​<m<a. What is the maximum possible value of a?
Answer Choices:
A. 10151​
B. 9950​
C. 10051​
D. 10152​
E. 2513​
Solution:
For 0<x≤100, the nearest lattice point directly above the line y=21​x+2 is (x,21​x+3) if x is even and (x,21​x+25​) if x is odd. The slope of the line that contains this point and (0,2) is 21​+x1​ if x is even and 21​+2x1​ if x is odd. The minimum value of the slope is 10051​ if x is even and 9950​ if x is odd. Therefore the line y=mx+2 contains no lattice point with 0<x≤100 for 21​<m<(B)9950​​.
The problems on this page are the property of the MAA's American Mathematics Competitions