What is the least number of square tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad?

Question

Accepted Answer

To determine the least number of square tiles required to pave the floor of a room measuring 15 meters 17 centimeters in length and 9 meters 2 centimeters in breadth, we need to follow these steps:

1. **Convert Measurements to Centimeters:**
   - Length of the room = 15 meters 17 centimeters = 1517 centimeters
   - Breadth of the room = 9 meters 2 centimeters = 902 centimeters

2. **Find the Greatest Common Divisor (GCD):**
   - To find the size of the largest square tile that can be used, we need to calculate the GCD of the room's length and breadth.
   - GCD of 1517 and 902 can be calculated using the Euclidean algorithm:
     - 1517 ÷ 902 = 1 remainder 615
     - 902 ÷ 615 = 1 remainder 287
     - 615 ÷ 287 = 2 remainder 41
     - 287 ÷ 41 = 7 remainder 0
   - Therefore, the GCD is 41 centimeters.

3. **Calculate the Number of Tiles:**
   - The side of each square tile is 41 centimeters.
   - Number of tiles along the length = 1517 ÷ 41 = 37
   - Number of tiles along the breadth = 902 ÷ 41 = 22
   - Total number of tiles = 37 × 22 = 814

Thus, the least number of square tiles required is 814, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (820):** This option suggests a higher number of tiles than necessary. Since the calculations show that 814 tiles are sufficient to cover the entire floor using the largest possible square tiles, 820 tiles would imply either smaller tiles or unnecessary overlap, which is not optimal.

- **Option 3 (840):** Similar to Option 2, this option suggests an even higher number of tiles. The calculation clearly shows that 814 tiles are enough, so 840 tiles would mean using smaller tiles or having excess tiles, which is not required.

- **Option 4 (841):** This option suggests a number of tiles that is very close to the correct answer but still incorrect. The precise calculation based on the GCD and the dimensions of the room confirms that 814 tiles are the minimum required, making 841 an overestimation.

In conclusion, Option 1 is the correct answer as it accurately reflects the minimum number of square tiles needed to cover the floor using the largest possible square tiles.