Determine the longest tape which can be used to measure exactly the lengths 7 m . 3 m 85 cm and 12 m 95 cm .

Answer

To determine the longest tape that can measure exactly the lengths of 7 m, 3 m 85 cm, and 12 m 95 cm, we need to follow these steps:

### Step 1: Convert all measurements to centimeters
1. **Convert 7 meters to centimeters**:
\[ 7 \text{ m} = 7 \times 100 = 700 \text{ cm} \]

2. **Convert 3 meters 85 centimeters to centimeters**:
\[ 3 \text{ m} 85 \text{ cm} = 3 \times 100 + 85 = 300 + 85 = 385 \text{ cm} \]

3. **Convert 12 meters 95 centimeters to centimeters**:
\[ 12 \text{ m} 95 \text{ cm} = 12 \times 100 + 95 = 1200 + 95 = 1295 \text{ cm} \]

### Step 2: Find the Highest Common Factor (HCF)
To find the longest tape that can measure all three lengths, we need to calculate the HCF of 700 cm, 385 cm, and 1295 cm.

#### Finding the factors:
1. **Factors of 700**:
- Divide by 2: \( 700 \div 2 = 350 \)
- Divide by 2: \( 350 \div 2 = 175 \)
- Divide by 5: \( 175 \div 5 = 35 \)
- Divide by 5: \( 35 \div 5 = 7 \)
- Divide by 7: \( 7 \div 7 = 1 \)

So, the prime factorization of 700 is:
\[ 700 = 2^2 \times 5^2 \times 7 \]

2. **Factors of 385**:
- Divide by 5: \( 385 \div 5 = 77 \)
- Divide by 7: \( 77 \div 7 = 11 \)
- Divide by 11: \( 11 \div 11 = 1 \)

So, the prime factorization of 385 is:
\[ 385 = 5^1 \times 7^1 \times 11^1 \]

3. **Factors of 1295**:
- Divide by 5: \( 1295 \div 5 = 259 \)
- Divide by 7: \( 259 \div 7 = 37 \)
- Divide by 37: \( 37 \div 37 = 1 \)

So, the prime factorization of 1295 is:
\[ 1295 = 5^1 \times 7^1 \times 37^1 \]

### Step 3: Identify common factors
The common prime factors in all three numbers are:
- \( 5^1 \)
- \( 7^1 \)

### Step 4: Calculate the HCF
The HCF is found by multiplying the lowest powers of the common prime factors:
\[ \text{HCF} = 5^1 \times 7^1 = 5 \times 7 = 35 \]

### Conclusion
The longest tape that can be used to measure exactly the lengths of 7 m, 3 m 85 cm, and 12 m 95 cm is **35 cm**.

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