Find the least number of square tiles by which the floor of a room of dimensions `16 . 58 xx 8 . 32 ` m can be covered completely ?

Options

A. 348644
B. 344864
C. 384644
D. None of these

Answer

To find the least number of square tiles required to cover the floor of a room with dimensions 16.58 m by 8.32 m, we can follow these steps:

### Step 1: Convert the dimensions to centimeters
Since the dimensions are given in meters, we will convert them to centimeters for easier calculations.
- 1 m = 100 cm
- Therefore,
- Length = 16.58 m = 16.58 × 100 = 1658 cm
- Width = 8.32 m = 8.32 × 100 = 832 cm

### Step 2: Find the HCF of the dimensions
To find the least number of square tiles, we need to determine the size of the square tile, which will be based on the highest common factor (HCF) of the two dimensions.

- We will find the HCF of 1658 and 832 using the Euclidean algorithm:
1. Divide 1658 by 832, which gives a quotient of 1 and a remainder of 826.
2. Now, divide 832 by 826, which gives a quotient of 1 and a remainder of 6.
3. Next, divide 826 by 6, which gives a quotient of 137 and a remainder of 4.
4. Then, divide 6 by 4, which gives a quotient of 1 and a remainder of 2.
5. Finally, divide 4 by 2, which gives a quotient of 2 and a remainder of 0.

The last non-zero remainder is 2, so the HCF of 1658 and 832 is 2 cm.

### Step 3: Calculate the area of the floor
The area of the floor can be calculated using the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
- Area = 1658 cm × 832 cm = 1,379,456 cm²

### Step 4: Calculate the area of one square tile
Since the size of each square tile is equal to the HCF, the side of the tile is 2 cm.
- Area of one tile = \( \text{side}^2 = 2 \times 2 = 4 \, \text{cm}^2 \)

### Step 5: Calculate the number of tiles required
To find the number of tiles needed, we divide the area of the floor by the area of one tile:
\[ \text{Number of tiles} = \frac{\text{Area of floor}}{\text{Area of one tile}} \]
- Number of tiles = \( \frac{1,379,456 \, \text{cm}^2}{4 \, \text{cm}^2} = 344,864 \)

### Conclusion
The least number of square tiles required to cover the floor completely is **344,864**.

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