Find the HCF and LCM of ` 8/9 , 10/29 and 16/81`

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the fractions \( \frac{8}{9} \), \( \frac{10}{29} \), and \( \frac{16}{81} \), we can use the following formulas:

1. **HCF of fractions**:
\[ HCF\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{HCF(a, c, e)}{LCM(b, d, f)} \]

2. **LCM of fractions**:
\[ LCM\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{LCM(a, c, e)}{HCF(b, d, f)} \]

### Step 1: Find the HCF of the numerators
The numerators are \( 8, 10, \) and \( 16 \).

- The factors of \( 8 \) are \( 1, 2, 4, 8 \).
- The factors of \( 10 \) are \( 1, 2, 5, 10 \).
- The factors of \( 16 \) are \( 1, 2, 4, 8, 16 \).

The common factors are \( 1 \) and \( 2 \). Thus, the HCF is \( 2 \).

### Step 2: Find the LCM of the denominators
The denominators are \( 9, 29, \) and \( 81 \).

- The factors of \( 9 \) are \( 1, 3, 9 \).
- The factors of \( 29 \) are \( 1, 29 \) (since 29 is a prime number).
- The factors of \( 81 \) are \( 1, 3, 9, 27, 81 \).

To find the LCM, we take the highest power of each prime factor:
- For \( 3 \): highest power is \( 3^4 = 81 \).
- For \( 29 \): highest power is \( 29^1 = 29 \).

Thus,
\[ LCM(9, 29, 81) = 81 \times 29 = 2349. \]

### Step 3: Calculate the HCF of the fractions
Using the formula:
\[ HCF\left(\frac{8}{9}, \frac{10}{29}, \frac{16}{81}\right) = \frac{HCF(8, 10, 16)}{LCM(9, 29, 81)} = \frac{2}{2349}. \]

### Step 4: Find the LCM of the numerators
Using the numerators \( 8, 10, 16 \):
- The LCM of \( 8, 10, 16 \):
- \( 8 = 2^3 \)
- \( 10 = 2^1 \times 5^1 \)
- \( 16 = 2^4 \)

Taking the highest power of each prime:
- For \( 2 \): highest power is \( 2^4 \).
- For \( 5 \): highest power is \( 5^1 \).

Thus,
\[ LCM(8, 10, 16) = 16 \times 5 = 80. \]

### Step 5: Calculate the HCF of the denominators
Using the denominators \( 9, 29, 81 \):
- Since \( 29 \) is a prime number and does not divide \( 9 \) or \( 81 \), the only common factor is \( 1 \).

### Step 6: Calculate the LCM of the fractions
Using the formula:
\[ LCM\left(\frac{8}{9}, \frac{10}{29}, \frac{16}{81}\right) = \frac{LCM(8, 10, 16)}{HCF(9, 29, 81)} = \frac{80}{1} = 80. \]

### Final Results
- HCF of the fractions: \( \frac{2}{2349} \)
- LCM of the fractions: \( 80 \)