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=21x+2 is (x,21x+3) if x is even and (x,21x+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<9950.
The problems on this page are the property of the MAA's American Mathematics Competitions