Find the HCF of `3/4, 5/6` and `6/7`

Options

A. `5/14`
B. `1/84`
C. `1/63`
D. `1/168`

Answer

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

### Step 1: Identify the Numerators and Denominators
We have the fractions:
- Numerators: 3, 5, 6
- Denominators: 4, 6, 7

### Step 2: Find the HCF of the Numerators
To find the HCF of the numerators (3, 5, 6):
- The prime factorization of 3 is \( 3^1 \)
- The prime factorization of 5 is \( 5^1 \)
- The prime factorization of 6 is \( 2^1 \times 3^1 \)

The only common factor among the numerators is 1. Thus, the HCF of the numerators is:
\[ \text{HCF}(3, 5, 6) = 1 \]

### Step 3: Find the LCM of the Denominators
To find the LCM of the denominators (4, 6, 7):
- The prime factorization of 4 is \( 2^2 \)
- The prime factorization of 6 is \( 2^1 \times 3^1 \)
- The prime factorization of 7 is \( 7^1 \)

To find the LCM, we take the highest power of each prime:
- For 2: \( 2^2 \)
- For 3: \( 3^1 \)
- For 7: \( 7^1 \)

Thus, the LCM is:
\[ \text{LCM}(4, 6, 7) = 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 \]
Calculating this:
\[ 4 \times 3 = 12 \quad \text{and} \quad 12 \times 7 = 84 \]
So, the LCM of the denominators is:
\[ \text{LCM}(4, 6, 7) = 84 \]

### Step 4: Form the HCF of the Fractions
Now, we can form the HCF of the fractions using the HCF of the numerators and the LCM of the denominators:
\[ \text{HCF}\left(\frac{3}{4}, \frac{5}{6}, \frac{6}{7}\right) = \frac{\text{HCF}(3, 5, 6)}{\text{LCM}(4, 6, 7)} = \frac{1}{84} \]

### Final Answer
Thus, the HCF of the fractions \( \frac{3}{4}, \frac{5}{6}, \frac{6}{7} \) is:
\[ \frac{1}{84} \]