A rectangular courtyard `1.12` m long and `0.84` m wide is to be paved exactly with square tiles , all of same size . The largest size of the which could be used be used for the purpose is `( n xx 4)` cm . Find the value of n .

Options

A. 6
B. 8
C. 7
D. None of these

Answer

To solve the problem step by step, we will find the largest size of the square tiles that can be used to pave the rectangular courtyard.

### Step 1: Convert the dimensions to centimeters
The dimensions of the courtyard are given in meters. We need to convert them to centimeters:
- Length = 1.12 m = 1.12 × 100 cm = 112 cm
- Width = 0.84 m = 0.84 × 100 cm = 84 cm

### Step 2: Find the HCF (Highest Common Factor) of the dimensions
To find the largest size of the square tile, we need to calculate the HCF of the length and width of the courtyard.

**Factorization of 112:**
- 112 ÷ 2 = 56
- 56 ÷ 2 = 28
- 28 ÷ 2 = 14
- 14 ÷ 2 = 7
- 7 ÷ 7 = 1

So, the prime factorization of 112 is:
\[ 112 = 2^4 \times 7 \]

**Factorization of 84:**
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1

So, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3 \times 7 \]

### Step 3: Identify the common factors
Now, we identify the common factors from the factorizations:
- Common factors: \( 2^2 \) (the minimum power of 2 from both) and \( 7 \)

### Step 4: Calculate the HCF
The HCF can be calculated by multiplying the common factors:
\[ \text{HCF} = 2^2 \times 7 = 4 \times 7 = 28 \]

### Step 5: Relate HCF to the tile size
According to the problem, the size of the square tile is given as \( n \times 4 \) cm. We found that the HCF is 28 cm. Therefore:
\[ n \times 4 = 28 \]

### Step 6: Solve for n
To find the value of \( n \):
\[ n = \frac{28}{4} = 7 \]

### Final Answer
The value of \( n \) is **7**.

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