HCF of `2/3, 4/5 and 6/7` is:

Options

A. `48/105`
B. `2/105`
C. `1/105`
D. `24/105`

Answer

To find the HCF (Highest Common Factor) of the fractions \( \frac{2}{3}, \frac{4}{5}, \frac{6}{7} \), we will use the formula:

\[ \text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} \]

### Step 1: Find the HCF of the numerators (2, 4, 6)

1. **Prime Factorization**:
- \( 2 = 2^1 \)
- \( 4 = 2^2 \)
- \( 6 = 2^1 \times 3^1 \)

2. **Identify Common Factors**:
- The common prime factor is \( 2 \).
- The lowest power of \( 2 \) in the factorizations is \( 2^1 \).

3. **Calculate HCF**:
- Therefore, \( \text{HCF}(2, 4, 6) = 2^1 = 2 \).

### Step 2: Find the LCM of the denominators (3, 5, 7)

1. **Prime Factorization**:
- \( 3 = 3^1 \)
- \( 5 = 5^1 \)
- \( 7 = 7^1 \)

2. **Identify All Factors**:
- Since all the numbers are prime, the LCM is simply the product of all the denominators.

3. **Calculate LCM**:
- \( \text{LCM}(3, 5, 7) = 3^1 \times 5^1 \times 7^1 = 3 \times 5 \times 7 = 105 \).

### Step 3: Calculate the HCF of the fractions

Using the values obtained:

\[ \text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{2}{105} \]

### Final Answer:
The HCF of \( \frac{2}{3}, \frac{4}{5}, \frac{6}{7} \) is \( \frac{2}{105} \).

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