The HCF and LCM of `13!` and `31!` are respectively :

Options

A. 12! and 32!
B. 13! and 31!
C. 26 and 403
D. can't be determined

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of \( 13! \) and \( 31! \), we can use the properties of factorials and the relationship between HCF and LCM.

### Step-by-Step Solution:

1. **Understanding Factorials**:
- The factorial of a number \( n \) (denoted as \( n! \)) is the product of all positive integers from 1 to \( n \).
- For example, \( 13! = 1 \times 2 \times 3 \times \ldots \times 13 \) and \( 31! = 1 \times 2 \times 3 \times \ldots \times 31 \).

2. **Identifying HCF**:
- The HCF of two numbers is the largest number that divides both of them.
- For factorials, the HCF of \( n! \) and \( m! \) (where \( n < m \)) is given by \( n! \).
- Therefore, \( \text{HCF}(13!, 31!) = 13! \).

3. **Identifying LCM**:
- The LCM of two numbers is the smallest number that is a multiple of both.
- The relationship between HCF and LCM can be expressed as:
\[ \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \]
- Applying this to our factorials:
\[ \text{HCF}(13!, 31!) \times \text{LCM}(13!, 31!) = 13! \times 31! \]
- We already found that \( \text{HCF}(13!, 31!) = 13! \).
- Therefore:
\[ 13! \times \text{LCM}(13!, 31!) = 13! \times 31! \]
- Dividing both sides by \( 13! \):
\[ \text{LCM}(13!, 31!) = \frac{31!}{1} = 31! \]

4. **Final Results**:
- The HCF of \( 13! \) and \( 31! \) is \( 13! \).
- The LCM of \( 13! \) and \( 31! \) is \( 31! \).

### Conclusion:
- The HCF of \( 13! \) and \( 31! \) is \( 13! \).
- The LCM of \( 13! \) and \( 31! \) is \( 31! \).