You are planning to lay square tiles on an entire 21125cm by 54925cm rectangular floor without any overlapping. If all the square tiles must be of the same size, what is the least possible number you can lay?
To find the least possible number of square tiles that can be laid on the rectangular floor without any overlapping, we need to determine the size of the square tiles.
First, we find the greatest common divisor (GCD) of the length and width of the rectangular floor. The GCD will be the size of each square tile that fits perfectly into the floor without leaving any leftover space.
To find the GCD, we can use the Euclidean algorithm, which is a method for finding the GCD of two numbers:
1. Start with the two numbers, 21125 and 54925.
2. Divide the larger number by the smaller number and find the remainder.
- 54925 ÷ 21125 = 2 remainder 12675
3. Replace the larger number with the smaller number and the smaller number with the remainder.
- New numbers: 21125 and 12675
4. Repeat steps 2 and 3 until the remainder becomes 0.
- 21125 ÷ 12675 = 1 remainder 8449
- 12675 ÷ 8449 = 1 remainder 4226
- 8449 ÷ 4226 = 2 remainder 19
- 4226 ÷ 19 = 222 remainder 8
- 19 ÷ 8 = 2 remainder 3
- 8 ÷ 3 = 2 remainder 2
- 3 ÷ 2 = 1 remainder 1
- 2 ÷ 1 = 2 remainder 0
5. The GCD is the last non-zero remainder, which is 1 in this case.
The GCD of 21125 and 54925 is 1. Hence, the size of each square tile that can be used is 1cm by 1cm.
Now, we can calculate the number of square tiles needed to cover the rectangular floor:
1. Divide the length of the rectangular floor by the size of each square tile.
- Number of tiles along the length = 54925 ÷ 1 = 54925 tiles
2. Divide the width of the rectangular floor by the size of each square tile.
- Number of tiles along the width = 21125 ÷ 1 = 21125 tiles
3. Multiply the number of tiles along the length by the number of tiles along the width to get the total number of tiles needed.
- Total number of tiles = 54925 tiles × 21125 tiles = 1,160,956,125 tiles
Therefore, the least possible number of square tiles required to cover the rectangular floor without any overlapping is 1,160,956,125 tiles, assuming each tile is 1cm by 1cm in size.