The HCF of two natural numbers m and n is 24 and their product is 552. How many sets of values of m and n are possible ?

Options

A. 1
B. 2
C. 4
D. No set of m and n is possible satisfying the given conditions

Answer

To solve the problem, we need to find the number of sets of values of two natural numbers \( m \) and \( n \) given that their HCF is 24 and their product is 552.

### Step-by-Step Solution:

1. **Understanding the relationship between HCF, LCM, and the product of two numbers**:
We know that the product of two numbers \( m \) and \( n \) can be expressed as:
\[ m \times n = \text{HCF}(m, n) \times \text{LCM}(m, n) \]

2. **Substituting the known values**:
Given that \( \text{HCF}(m, n) = 24 \) and \( m \times n = 552 \), we can substitute these values into the equation:
\[ 552 = 24 \times \text{LCM}(m, n) \]

3. **Calculating the LCM**:
To find the LCM, we rearrange the equation:
\[ \text{LCM}(m, n) = \frac{552}{24} \]
Now, we perform the division:
\[ \text{LCM}(m, n) = 23 \]

4. **Finding the possible values of \( m \) and \( n \)**:
We know that for two numbers \( m \) and \( n \) with HCF = 24, we can express them as:
\[ m = 24a \quad \text{and} \quad n = 24b \]
where \( a \) and \( b \) are coprime (i.e., \( \text{HCF}(a, b) = 1 \)).

5. **Substituting into the product equation**:
Now substituting \( m \) and \( n \) into the product equation:
\[ m \times n = (24a)(24b) = 552 \]
Simplifying gives:
\[ 576ab = 552 \]
Dividing both sides by 24:
\[ 24ab = 23 \]

6. **Analyzing the equation**:
The equation \( 24ab = 23 \) implies that \( ab = \frac{23}{24} \). Since \( a \) and \( b \) must be natural numbers, \( ab \) cannot equal a fraction. Therefore, there are no natural number solutions for \( a \) and \( b \).

7. **Conclusion**:
Since there are no valid pairs of \( (m, n) \) that satisfy the conditions given, we conclude that there are no sets of values of \( m \) and \( n \) possible.

### Final Answer:
There are no sets of values of \( m \) and \( n \) possible.