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 ?

Options

A. 840
B. 841
C. 820
D. 814

Answer

To find the least number of square tiles required to pave the floor of a room measuring 15 m 17 cm long and 9 m 2 cm broad, we can follow these steps:

### Step 1: Convert dimensions to centimeters
First, we need to convert the dimensions of the room from meters and centimeters to just centimeters.

- Length:
\[ 15 \text{ m} = 15 \times 100 = 1500 \text{ cm} \]
\[ 1500 \text{ cm} + 17 \text{ cm} = 1517 \text{ cm} \]

- Breadth:
\[ 9 \text{ m} = 9 \times 100 = 900 \text{ cm} \]
\[ 900 \text{ cm} + 2 \text{ cm} = 902 \text{ cm} \]

### Step 2: Calculate the area of the room
Next, we calculate the area of the room using the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
\[ \text{Area} = 1517 \text{ cm} \times 902 \text{ cm} \]

### Step 3: Calculate the area of the tiles
We need to find the size of the square tiles. To do this, we will find the highest common factor (HCF) of the dimensions of the room (1517 cm and 902 cm).

### Step 4: Find the HCF
Using the Euclidean algorithm, we can find the HCF of 1517 and 902.

1. Divide 1517 by 902, which gives a quotient of 1 and a remainder of 615.
2. Next, divide 902 by 615, which gives a quotient of 1 and a remainder of 287.
3. Then, divide 615 by 287, which gives a quotient of 2 and a remainder of 41.
4. Finally, divide 287 by 41, which gives a quotient of 7 and a remainder of 0.

Since the last non-zero remainder is 41, the HCF is 41 cm.

### Step 5: Calculate the area of one tile
The area of one square tile is given by:
\[ \text{Area of tile} = \text{side} \times \text{side} = 41 \text{ cm} \times 41 \text{ cm} = 1681 \text{ cm}^2 \]

### Step 6: Calculate the total area of the room
Now we can calculate the total area of the room:
\[ \text{Area of room} = 1517 \text{ cm} \times 902 \text{ cm} = 1362444 \text{ cm}^2 \]

### Step 7: Calculate the number of tiles required
Finally, we can find the number of tiles required by dividing the area of the room by the area of one tile:
\[ \text{Number of tiles} = \frac{\text{Area of room}}{\text{Area of tile}} = \frac{1362444 \text{ cm}^2}{1681 \text{ cm}^2} \approx 810 \]

### Conclusion
Thus, the least number of square tiles required to pave the floor of the room is **814 tiles**.

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