You have a rectangular sheet of metal of dimensions 60 inches by 84 inches. You want to cut this metal into smaller identical squares. What is the largest size square you would need to ensure there is no metal left over?
The squares have equal length and width, let's call this $x$. This $x$ must divide 60 and 84. The largest such $x$ is the greatest common divisor of 60 and 84.
The factors of 60 are $\{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}$.
The factors of 84 are $\{1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84\}$.
The greatest common divisor is 12.
So the largest size square is 12 inches.