Three cloth merchants have total length of cloth in 57 meters, 69 meters and 93 meters. Find the greatest measure (meter) which can measure the clothes completely (exactly)

Options

A. 6
B. 1
C. 7
D. None

Answer

To find the greatest measure (in meters) that can measure the lengths of cloth (57 meters, 69 meters, and 93 meters) completely, we need to determine the Highest Common Factor (HCF) of these three lengths. Here’s a step-by-step solution:

### Step 1: List the lengths of cloth
The lengths of cloth are:
- Cloth 1: 57 meters
- Cloth 2: 69 meters
- Cloth 3: 93 meters

### Step 2: Find the prime factorization of each length
1. **Prime Factorization of 57**:
- 57 can be divided by 3 (57 ÷ 3 = 19).
- Therefore, the prime factorization of 57 is:
\[ 57 = 3 \times 19 \]

2. **Prime Factorization of 69**:
- 69 can also be divided by 3 (69 ÷ 3 = 23).
- Therefore, the prime factorization of 69 is:
\[ 69 = 3 \times 23 \]

3. **Prime Factorization of 93**:
- 93 can again be divided by 3 (93 ÷ 3 = 31).
- Therefore, the prime factorization of 93 is:
\[ 93 = 3 \times 31 \]

### Step 3: Identify the common factors
From the prime factorizations:
- 57 = 3 × 19
- 69 = 3 × 23
- 93 = 3 × 31

The only common prime factor among all three numbers is **3**.

### Step 4: Determine the HCF
Since 3 is the only common factor, the HCF of 57, 69, and 93 is:
\[ \text{HCF} = 3 \]

### Step 5: Conclusion
The greatest measure (in meters) that can measure the lengths of cloth completely is:
\[ \text{Answer} = 3 \text{ meters} \]

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