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