If `l=`LCM of `8!, 10!` and `12!` and `h =` HCF of of `8!, 10!` and `12!` then `l/h` is equal to a)132 b)11800 c)11880 d)None of these

Options

A. 132
B. 11800
C. 11880
D. None of these

Explanation

To solve the problem, we need to find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of \(8!\), \(10!\), and \(12!\), and then calculate the ratio \( \frac{L}{H} \).

### Step-by-Step Solution:

1. **Understanding Factorials**:
- Recall that \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\).
- Thus, we can express \(10!\) and \(12!\) in terms of \(8!\):
\[ 10! = 10 \times 9 \times 8! \]
\[ 12! = 12 \times 11 \times 10 \times 9 \times 8! \]

2. **Finding the LCM**:
- The LCM of several numbers is the product of the highest powers of all prime factors present in those numbers.
- The prime factorization of \(10!\) and \(12!\) can be derived from \(8!\):
- \(10! = 10 \times 9 \times 8! = (2 \times 5) \times (3^2) \times 8!\)
- \(12! = 12 \times 11 \times 10 \times 9 \times 8! = (2^2 \times 3) \times 11 \times (2 \times 5) \times (3^2) \times 8!\)

3. **Identifying Prime Factors**:
- The prime factors from \(8!\), \(10!\), and \(12!\) include \(2\), \(3\), \(5\), and \(11\).
- The highest powers of these factors:
- For \(2\): The highest power is \(2^3\) (from \(12!\)).
- For \(3\): The highest power is \(3^3\) (from \(10!\) and \(12!\)).
- For \(5\): The highest power is \(5^1\) (from \(10!\)).
- For \(11\): The highest power is \(11^1\) (from \(12!\)).

4. **Calculating the LCM**:
- Thus, the LCM can be expressed as:
\[ L = 2^3 \times 3^3 \times 5^1 \times 11^1 \times 8! \]

5. **Finding the HCF**:
- The HCF is the product of the lowest powers of all prime factors present in \(8!\), \(10!\), and \(12!\).
- The only common factor among all three is \(8!\):
\[ H = 8! \]

6. **Calculating \( \frac{L}{H} \)**:
- Now, we can calculate:
\[ \frac{L}{H} = \frac{2^3 \times 3^3 \times 5 \times 11 \times 8!}{8!} = 2^3 \times 3^3 \times 5 \times 11 \]

7. **Calculating the Numerical Value**:
- Now we calculate:
\[ 2^3 = 8, \quad 3^3 = 27, \quad 5 = 5, \quad 11 = 11 \]
- Therefore,
\[ 8 \times 27 \times 5 \times 11 \]
- First, calculate \(8 \times 27 = 216\).
- Next, calculate \(216 \times 5 = 1080\).
- Finally, calculate \(1080 \times 11 = 11880\).

### Final Answer:
\[ \frac{L}{H} = 11880 \]