What is the length of the longest tape which can be used to measure exactly 1 m 75 cm, 4 m 50 cm and 6 m 50 cm?

Options

A. 25 cm
B. 50 cm
C. 55 cm
D. 75 cm

Answer

To find the length of the longest tape that can measure exactly 1 m 75 cm, 4 m 50 cm, and 6 m 50 cm, we need to follow these steps:

### Step 1: Convert all measurements to centimeters
1. **1 m 75 cm** = \(1 \times 100 + 75 = 175\) cm
2. **4 m 50 cm** = \(4 \times 100 + 50 = 450\) cm
3. **6 m 50 cm** = \(6 \times 100 + 50 = 650\) cm

### Step 2: Write down the converted measurements
- The measurements in centimeters are:
- 175 cm
- 450 cm
- 650 cm

### Step 3: Find the prime factorization of each measurement
1. **Prime factorization of 175**:
- \(175 = 5 \times 35\)
- \(35 = 5 \times 7\)
- So, \(175 = 5^2 \times 7\)

2. **Prime factorization of 450**:
- \(450 = 45 \times 10\)
- \(45 = 9 \times 5\) and \(9 = 3^2\), so \(45 = 3^2 \times 5\)
- \(10 = 2 \times 5\)
- So, \(450 = 2 \times 3^2 \times 5^2\)

3. **Prime factorization of 650**:
- \(650 = 65 \times 10\)
- \(65 = 5 \times 13\) and \(10 = 2 \times 5\)
- So, \(650 = 2 \times 5^2 \times 13\)

### Step 4: List the prime factors
- **175**: \(5^2 \times 7\)
- **450**: \(2^1 \times 3^2 \times 5^2\)
- **650**: \(2^1 \times 5^2 \times 13\)

### Step 5: Identify the common prime factors
- The common prime factor among all three numbers is \(5^2\).

### Step 6: Calculate the Highest Common Factor (HCF)
- The HCF is \(5^2 = 25\).

### Conclusion
The length of the longest tape that can measure exactly 1 m 75 cm, 4 m 50 cm, and 6 m 50 cm is **25 cm**.

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