Thoughts About an Ugly Bathroom Floor
A rectangular floor is composed of congruent square tiles, 81 tiles along one side and 63 tiles along the other. If a straight line is drawn diagonally across the floor between two opposite corners, how many tiles will the line cross?
Now, generalize: Suppose the rectangular floor was again composed of congruent square tiles, but now m tiles along one side and n tiles along the other. If a straight line is drawn diagonally across the floor between two opposite corners, how many tiles will the line cross?
Hint: First, you could create a version of the floor using graph paper....but there has to be a better way.
Think: reduce the size of the problem and look for a pattern.
Solution Commentary: When looking for a pattern, try graph paper versions of tiled floors involving these dimensions: 2 x 2, 2 x 4, 4 x 4, 4 x 5, 4 x 6, 8 x 9, 8 x 10, and 8 x 12.
Look at your obtained number for each case...and focus on the properties/relationships of the floor size. For example, what is different about the results for the 4 x 5 floor and the 4 x 6 floor? Also, how is the 2 x 4 floor 2 x 4 connected to the 4 x 8 floor?
If you have focused on these questions carefully, you probably have not only solved the 63 x 81 floor problem, but also the m x n floor problem.
If not, think about things like gcd's and "root floors."