Problem:
A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.)
Answer Choices:
A. 2
B. 4
C. 6
D. 16
E. 46
Solution:
Since 48−3281−17=1588, the lattice point (x,y) is on the line segment if and only if
x and y are integers ,3≤x≤48, and y=17+1588(x−3).
But y is an integer if and only if x−3 is a multiple of 15. (Why?) The 4 lattice points are, therefore, (3,17),(18,105),(33,193) and (48,281).