Example 12 A steel plate, 1丈 3尺 5寸 long and 1丈 5寸 wide. Now it is to be cut into equally sized squares, the squares must be as large as possible, and no steel plate should be left over. Find the side length of the square.

Note: In traditional Chinese units, 1丈 = 10尺, and 1尺 = 10寸.
Solution: Since we want the largest square, we need to find the largest side length of the square. To find the largest side length of the square, we need to find the greatest common divisor (GCD) of 135 inches and 105 inches. Since
$$135=3^{3} \times 5, \quad 105=3 \times 5 \times 7$$

we have $(135,105)=15$.
Answer: The side length of the square is 1 foot 5 inches.