What will be the HCF of `(2 xx 3 xx 7 xx 9),(2 xx3 xx 9 xx 11 ) and `(2 x 3 x 4 x 5)`?

Options

A. `2xx3xx7`
B. `2xx3xx9`
C. `2xx3`
D. `2xx7xx9xx11`

Answer

To find the HCF (Highest Common Factor) of the numbers given, we will follow these steps:

1. **Identify the numbers**:
- The first number is \(2 \times 3 \times 7 \times 9\).
- The second number is \(2 \times 3 \times 9 \times 11\).
- The third number is \(2 \times 3 \times 4 \times 5\).

2. **Factorization of each number**:
- For the first number:
\[ 2 \times 3 \times 7 \times 9 = 2 \times 3 \times 7 \times (3^2) = 2 \times 3^3 \times 7 \]
- For the second number:
\[ 2 \times 3 \times 9 \times 11 = 2 \times 3 \times (3^2) \times 11 = 2 \times 3^3 \times 11 \]
- For the third number:
\[ 2 \times 3 \times 4 \times 5 = 2 \times 3 \times (2^2) \times 5 = 2^3 \times 3 \times 5 \]

3. **List the prime factors**:
- First number: \(2^1, 3^3, 7^1\)
- Second number: \(2^1, 3^3, 11^1\)
- Third number: \(2^3, 3^1, 5^1\)

4. **Find the common prime factors**:
- The common prime factors among all three numbers are:
- For \(2\): The minimum power is \(2^1\) (from the first and second numbers).
- For \(3\): The minimum power is \(3^1\) (from the third number).

5. **Calculate the HCF**:
- The HCF is obtained by multiplying the common prime factors with their minimum powers:
\[ HCF = 2^1 \times 3^1 = 2 \times 3 = 6 \]

Thus, the HCF of \(2 \times 3 \times 7 \times 9\), \(2 \times 3 \times 9 \times 11\), and \(2 \times 3 \times 4 \times 5\) is **6**.