The product of the LCM and the HCF of two numbers is 24. if the difference of the numbers is 2 , then the greater of the number is

Options

A. 3
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to find the greater of two numbers given that the product of their LCM (Least Common Multiple) and HCF (Highest Common Factor) is 24, and the difference between the two numbers is 2.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We know that for any two numbers \( a \) and \( b \):
\[ \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b \]
Given that \( \text{LCM} \times \text{HCF} = 24 \), we can express this as:
\[ a \times b = 24 \]

2. **Setting Up the Variables**:
Let the greater number be \( x \) and the smaller number be \( y \). According to the problem, the difference between the two numbers is:
\[ x - y = 2 \]
From this, we can express \( y \) in terms of \( x \):
\[ y = x - 2 \]

3. **Substituting into the Product Equation**:
Now substituting \( y \) into the product equation:
\[ x \times (x - 2) = 24 \]
Expanding this gives:
\[ x^2 - 2x = 24 \]

4. **Rearranging the Equation**:
Rearranging the equation to standard quadratic form:
\[ x^2 - 2x - 24 = 0 \]

5. **Factoring the Quadratic**:
We need to factor the quadratic equation. We look for two numbers that multiply to \(-24\) and add to \(-2\). The numbers \(-6\) and \(4\) work:
\[ (x - 6)(x + 4) = 0 \]

6. **Finding the Roots**:
Setting each factor to zero gives us:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]

7. **Selecting the Valid Solution**:
Since \( x \) represents a number, we discard \( x = -4 \) as it is not valid. Thus, we have:
\[ x = 6 \]

8. **Conclusion**:
The greater number is \( x = 6 \).

### Final Answer:
The greater of the two numbers is **6**.