The HCF and LCM of two numbers are 12 and 924 respectively. Then the numbers of such pairs is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem of finding the number of pairs of two numbers whose HCF is 12 and LCM is 924, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding HCF and LCM**:
- The HCF (Highest Common Factor) of two numbers is the largest number that divides both of them.
- The LCM (Least Common Multiple) of two numbers is the smallest number that is a multiple of both.

2. **Using the Relationship Between HCF and LCM**:
- The relationship between HCF, LCM, and the two numbers \( a \) and \( b \) can be expressed as:
\[ HCF(a, b) \times LCM(a, b) = a \times b \]
- Here, we know that \( HCF = 12 \) and \( LCM = 924 \).

3. **Expressing the Numbers**:
- Since the HCF is 12, we can express the two numbers as:
\[ a = 12x \quad \text{and} \quad b = 12y \]
- Here, \( x \) and \( y \) are coprime (they have no common factors other than 1).

4. **Substituting into the Relationship**:
- Substituting \( a \) and \( b \) into the relationship gives:
\[ 12 \times 924 = (12x) \times (12y) \]
- Simplifying this, we get:
\[ 12 \times 924 = 144xy \]
- Dividing both sides by 12:
\[ 924 = 12xy \]
- Dividing by 12 again:
\[ xy = \frac{924}{12} = 77 \]

5. **Finding Pairs of \( (x, y) \)**:
- Now, we need to find pairs of integers \( (x, y) \) such that \( xy = 77 \).
- The pairs of factors of 77 are:
- \( (1, 77) \)
- \( (7, 11) \)

6. **Counting Unique Pairs**:
- The pairs \( (1, 77) \) and \( (7, 11) \) give us the following combinations:
- \( (12 \times 1, 12 \times 77) \) → \( (12, 924) \)
- \( (12 \times 7, 12 \times 11) \) → \( (84, 132) \)
- Each pair represents a valid combination of numbers.

7. **Conclusion**:
- Therefore, the total number of unique pairs of numbers \( (a, b) \) such that their HCF is 12 and their LCM is 924 is **2**.

### Final Answer:
The number of such pairs is **2**.