The largest possible length of a tape which can measure 525 cm, 1050 cm and 1155 cm length of cloths in a minium number of attempts without measuring the length of a cloth in a fraction of the tape's length

Options

A. 25
B. 105
C. 75
D. None of these

Answer

To find the largest possible length of a tape that can measure 525 cm, 1050 cm, and 1155 cm in a minimum number of attempts without measuring in fractions, we need to determine the highest common factor (HCF) of these three lengths. Here’s how to solve the problem step by step:

### Step 1: Prime Factorization of Each Length
We will start by finding the prime factorization of each of the three lengths.

1. **For 525:**
- Divide by 5: \( 525 \div 5 = 105 \)
- Divide 105 by 5: \( 105 \div 5 = 21 \)
- Divide 21 by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 525 is:
\[ 525 = 5^2 \times 3^1 \times 7^1 \]

2. **For 1050:**
- Divide by 5: \( 1050 \div 5 = 210 \)
- Divide 210 by 5: \( 210 \div 5 = 42 \)
- Divide 42 by 2: \( 42 \div 2 = 21 \)
- Divide 21 by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 1050 is:
\[ 1050 = 5^2 \times 2^1 \times 3^1 \times 7^1 \]

3. **For 1155:**
- Divide by 5: \( 1155 \div 5 = 231 \)
- Divide 231 by 3: \( 231 \div 3 = 77 \)
- Divide 77 by 7: \( 77 \div 7 = 11 \)
- 11 is a prime number.
- Therefore, the prime factorization of 1155 is:
\[ 1155 = 5^1 \times 3^1 \times 7^1 \times 11^1 \]

### Step 2: Identify Common Factors
Next, we identify the common prime factors from the factorizations:

- **Common factors:**
- \( 5 \) appears in all three factorizations.
- \( 3 \) appears in all three factorizations.
- \( 7 \) appears in all three factorizations.

### Step 3: Calculate the HCF
To find the HCF, we take the lowest power of each common prime factor:

- For \( 5 \): The lowest power is \( 5^1 \).
- For \( 3 \): The lowest power is \( 3^1 \).
- For \( 7 \): The lowest power is \( 7^1 \).

Now, we multiply these together to find the HCF:
\[ HCF = 5^1 \times 3^1 \times 7^1 = 5 \times 3 \times 7 = 105 \]

### Conclusion
The largest possible length of the tape that can measure 525 cm, 1050 cm, and 1155 cm is **105 cm**.