What will be the minimum number of squaretles required to cover a floor are a of dimensions 6m 20cm and 7m 40cm?

Options

A. 31
B. 37
C. 4588
D. 1147

Answer

To solve the problem of finding the minimum number of square tiles required to cover a floor area with dimensions of 6m 20cm and 7m 40cm, we will follow these steps:

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

- **6 meters 20 centimeters** = \(6 \times 100 + 20 = 620\) cm
- **7 meters 40 centimeters** = \(7 \times 100 + 40 = 740\) cm

### Step 2: Find the prime factorization of both dimensions
Next, we will find the prime factorization of both dimensions.

- For **620**:
- Divide by 2: \(620 \div 2 = 310\)
- Divide by 2: \(310 \div 2 = 155\)
- Divide by 5: \(155 \div 5 = 31\)
- 31 is a prime number.

So, the prime factorization of 620 is \(2^2 \times 5^1 \times 31^1\).

- For **740**:
- Divide by 2: \(740 \div 2 = 370\)
- Divide by 2: \(370 \div 2 = 185\)
- Divide by 5: \(185 \div 5 = 37\)
- 37 is a prime number.

So, the prime factorization of 740 is \(2^2 \times 5^1 \times 37^1\).

### Step 3: Find the Highest Common Factor (HCF)
Now, we find the HCF of the two numbers using their prime factorizations.

- The common prime factors are \(2^2\) and \(5^1\).
- Therefore, the HCF is \(2^2 \times 5^1 = 4 \times 5 = 20\).

### Step 4: Calculate the area of the floor
Next, we calculate the area of the floor.

- Area = Length × Width = \(620 \, \text{cm} \times 740 \, \text{cm}\)

### Step 5: Calculate the number of square tiles
To find the minimum number of square tiles required, we divide the area of the floor by the area of one square tile (which is the square of the HCF).

- Area of one square tile = \(HCF^2 = 20^2 = 400 \, \text{cm}^2\)
- Total area of the floor = \(620 \times 740 = 458800 \, \text{cm}^2\)
- Number of square tiles = \(\frac{\text{Total Area}}{\text{Area of one tile}} = \frac{458800}{400}\)

Now, calculating this gives:

- Number of square tiles = \(1147\)

### Conclusion
The minimum number of square tiles required to cover the floor area is **1147**.

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