The HCF of `(2)/(3),(8)/(9),(10)/(27),(32)/(81)` is:

Options

A. `(160)/(81)`
B. `(160)/(3)`
C. `(2)/(3)`
D. `(2)/(81)`

Answer

To find the HCF (Highest Common Factor) of the fractions \( \frac{2}{3}, \frac{8}{9}, \frac{10}{27}, \frac{32}{81} \), we can use the formula:

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

### Step 1: Identify the numerators and denominators
The numerators are: 2, 8, 10, 32
The denominators are: 3, 9, 27, 81

### Step 2: Calculate the HCF of the numerators
To find the HCF of the numerators (2, 8, 10, 32):

- The factors of 2: 1, 2
- The factors of 8: 1, 2, 4, 8
- The factors of 10: 1, 2, 5, 10
- The factors of 32: 1, 2, 4, 8, 16, 32

The common factors are 1 and 2. Thus, the HCF of 2, 8, 10, and 32 is **2**.

### Step 3: Calculate the LCM of the denominators
To find the LCM of the denominators (3, 9, 27, 81):

- The prime factorization of 3: \( 3^1 \)
- The prime factorization of 9: \( 3^2 \)
- The prime factorization of 27: \( 3^3 \)
- The prime factorization of 81: \( 3^4 \)

The LCM is found by taking the highest power of each prime factor. Here, the highest power of 3 is \( 3^4 \).

Thus, the LCM of 3, 9, 27, and 81 is **81**.

### Step 4: Calculate the HCF of the fractions
Now, we can substitute the HCF of the numerators and the LCM of the denominators into the formula:

\[ \text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{2}{81} \]

### Final Answer:
The HCF of the fractions \( \frac{2}{3}, \frac{8}{9}, \frac{10}{27}, \frac{32}{81} \) is \( \frac{2}{81} \).

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