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

Options

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

Answer

To solve the problem, we need to find the number of pairs of integers (x, y) such that the highest common factor (HCF) is 12 and the least common multiple (LCM) is 924.

### Step-by-Step Solution:

1. **Understanding HCF and LCM**:
The relationship between the HCF and LCM of two numbers \( a \) and \( b \) is given by:
\[ \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \]
Given that HCF = 12 and LCM = 924, we can express this as:
\[ 12 \times 924 = a \times b \]

2. **Expressing the Numbers**:
Since the HCF of the two numbers is 12, we can express the two numbers as:
\[ a = 12x \quad \text{and} \quad b = 12y \]
where \( x \) and \( y \) are coprime (i.e., HCF(x, y) = 1).

3. **Substituting into the LCM Formula**:
The LCM of \( a \) and \( b \) can be expressed as:
\[ \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)} = \frac{(12x) \times (12y)}{12} = 12xy \]
Setting this equal to the given LCM:
\[ 12xy = 924 \]

4. **Simplifying the Equation**:
Dividing both sides by 12 gives:
\[ xy = \frac{924}{12} = 77 \]

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

We check the HCF for each pair:
- For (1, 77), HCF(1, 77) = 1 (valid pair)
- For (7, 11), HCF(7, 11) = 1 (valid pair)

6. **Counting the Valid Pairs**:
The valid pairs (x, y) are (1, 77) and (7, 11). Each pair can be arranged in two ways (x, y) and (y, x), so we have:
- (1, 77) and (77, 1)
- (7, 11) and (11, 7)

Thus, the total number of pairs is:
\[ 2 + 2 = 4 \]

### Final Answer:
The number of such pairs is **4**.