Problem:
How many lines in a three-dimensional rectangular coordinate system pass through four distinct points of the form (), where and are positive integers not exceeding
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Consider the smallest cube containing all the lattice points , in a three dimensional Cartesian coordinate system. There are main diagonals. There are diagonal lines parallel to a coordinate plane: in each of four planes parallel to each of the three coordinate planes. There are lines parallel to a coordinate axis: in each of the three directions. Therefore, there are lines.