The product of LCM and HCF of `4/5 , 6/7 and 7/9` is _______.

Options

A. `4/5`
B. `3/7`
C. `8/15`
D. `5/7`

Answer

To find the product of the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the fractions \( \frac{4}{5}, \frac{6}{7}, \frac{7}{9} \), we can follow these steps:

### Step 1: Identify the Numerators and Denominators
The fractions given are:
- \( \frac{4}{5} \)
- \( \frac{6}{7} \)
- \( \frac{7}{9} \)

From these fractions, we identify:
- Numerators: \( 4, 6, 7 \)
- Denominators: \( 5, 7, 9 \)

### Step 2: Calculate the LCM of the Numerators
To find the LCM of \( 4, 6, \) and \( 7 \):
- The prime factorization is:
- \( 4 = 2^2 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 7 = 7^1 \)

The LCM is found by taking the highest power of each prime:
- LCM = \( 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 = 84 \)

### Step 3: Calculate the HCF of the Denominators
To find the HCF of \( 5, 7, \) and \( 9 \):
- The prime factorization is:
- \( 5 = 5^1 \)
- \( 7 = 7^1 \)
- \( 9 = 3^2 \)

Since there are no common prime factors, the HCF is:
- HCF = \( 1 \)

### Step 4: Calculate the Product of LCM and HCF
Now, we can calculate the product of the LCM of the numerators and the HCF of the denominators:
- Product = LCM of numerators \( \times \) HCF of denominators
- Product = \( 84 \times 1 = 84 \)

### Final Answer
The product of the LCM and HCF of \( \frac{4}{5}, \frac{6}{7}, \frac{7}{9} \) is \( 84 \).

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