2024 AMC 8 Problems/Problem 23
Problem
Rodrigo is drawing lines on the coordinate plane, and counting how many unit squares they go through. He draws a line with endpoints and
How many unit squares does this segment go through?
Solution 1
Let be the number of cells the line segment from
to
passes through. The problem is then equivalent to finding
Sometimes the segment passes through lattice points in between the endpoints, which happens
times. This partitions the segment into
congruent pieces that pass through
cells, which means the answer is
Note that a new square is entered when the lines pass through one of the lines in the coordinate grid, which for
happens
times. Because
and
are relatively prime, no lattice point except for the endpoints intersects the line segment from
to
This means that including the first cell closest to
The segment passes through
cells. Thus, the answer is
~BS2012