The dimensions of a hall are 6048 cm `xx` 2440 cm. What is the minimum number of identical square shaped tiles is required to furnish the hall ?

Options

A. 230580
B. 1061
C. 1240
D. 2320

Explanation

To find the minimum number of identical square-shaped tiles required to furnish the hall with dimensions 6048 cm x 2440 cm, we need to follow these steps:

### Step 1: Find the dimensions of the hall
The dimensions of the hall are given as:
- Length = 6048 cm
- Width = 2440 cm

### Step 2: Find the Highest Common Factor (HCF)
To determine the size of the square tiles, we need to find the HCF of the two dimensions. The HCF will give us the side length of the largest square tile that can fit perfectly into both dimensions.

#### Finding the HCF of 6048 and 2440:
1. **Prime factorization of 6048**:
- 6048 = 2 × 3024
- 3024 = 2 × 1512
- 1512 = 2 × 756
- 756 = 2 × 378
- 378 = 2 × 189
- 189 = 3 × 63
- 63 = 3 × 21
- 21 = 3 × 7
- So, the prime factorization of 6048 is:
- \( 6048 = 2^4 × 3^3 × 7^1 \)

2. **Prime factorization of 2440**:
- 2440 = 2 × 1220
- 1220 = 2 × 610
- 610 = 2 × 305
- 305 = 5 × 61
- So, the prime factorization of 2440 is:
- \( 2440 = 2^3 × 5^1 × 61^1 \)

3. **Finding the HCF**:
- The common prime factor is 2.
- The lowest power of 2 in both factorizations is \( 2^3 \).
- Therefore, HCF = \( 2^3 = 8 \) cm.

### Step 3: Calculate the area of the hall
The area of the hall can be calculated as:
\[ \text{Area of the hall} = \text{Length} \times \text{Width} = 6048 \, \text{cm} \times 2440 \, \text{cm} \]

Calculating this gives:
\[ \text{Area of the hall} = 6048 \times 2440 = 14761920 \, \text{cm}^2 \]

### Step 4: Calculate the area of one tile
The area of one square tile with side length equal to the HCF (8 cm) is:
\[ \text{Area of one tile} = \text{side}^2 = 8 \, \text{cm} \times 8 \, \text{cm} = 64 \, \text{cm}^2 \]

### Step 5: Calculate the number of tiles required
To find the number of tiles required, divide the area of the hall by the area of one tile:
\[ \text{Number of tiles} = \frac{\text{Area of the hall}}{\text{Area of one tile}} = \frac{14761920 \, \text{cm}^2}{64 \, \text{cm}^2} \]

Calculating this gives:
\[ \text{Number of tiles} = 230400 \]

### Final Answer
The minimum number of identical square-shaped tiles required to furnish the hall is **230400**.

---