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 follow these steps:

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

The common factor among all these numbers is 2. Therefore, the HCF of the numerators is:
\[ \text{HCF} = 2 \]

### Step 3: Calculate the LCM of the Denominators
To find the LCM of the denominators (3, 9, 27, 81):
- The prime factorization of 3 is \(3^1\)
- The prime factorization of 9 is \(3^2\)
- The prime factorization of 27 is \(3^3\)
- The prime factorization of 81 is \(3^4\)

To find the LCM, we take the highest power of each prime factor:
\[ \text{LCM} = 3^4 = 81 \]

### Step 4: Combine HCF and LCM
According to the relationship between HCF and LCM of fractions:
\[ \text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} \]
Substituting the values we found:
\[ \text{HCF} = \frac{2}{81} \]

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