The product of the LCM and HCF of two numbers is 24.The difference of the two numbers is 2.Find the numbers?

Options

A. 8 and 6
B. 8 and 10
C. 2 and 4
D. 6 and 4

Answer

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

Let's denote the two numbers as \( X \) and \( Y \).

### Step 1: Set up the equations
From the problem, we know:
1. \( \text{LCM}(X, Y) \times \text{HCF}(X, Y) = X \times Y = 24 \) (Equation 1)
2. The difference between the two numbers is \( |X - Y| = 2 \) (Equation 2)

### Step 2: Express one variable in terms of the other
From Equation 2, we can express \( X \) in terms of \( Y \):
- If \( X - Y = 2 \), then \( X = Y + 2 \).

### Step 3: Substitute into the product equation
Now, substitute \( X = Y + 2 \) into Equation 1:
\[ (Y + 2) \times Y = 24 \]
Expanding this gives:
\[ Y^2 + 2Y = 24 \]

### Step 4: Rearrange to form a quadratic equation
Rearranging the equation gives:
\[ Y^2 + 2Y - 24 = 0 \]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ Y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = 2, c = -24 \).

Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \times 1 \times (-24) = 4 + 96 = 100 \]

Now, substituting back into the quadratic formula:
\[ Y = \frac{-2 \pm \sqrt{100}}{2 \times 1} = \frac{-2 \pm 10}{2} \]

This gives us two possible solutions for \( Y \):
1. \( Y = \frac{8}{2} = 4 \)
2. \( Y = \frac{-12}{2} = -6 \) (not valid since numbers cannot be negative)

### Step 6: Find the corresponding value of X
Now, substituting \( Y = 4 \) back into the equation for \( X \):
\[ X = Y + 2 = 4 + 2 = 6 \]

### Conclusion
The two numbers are \( X = 6 \) and \( Y = 4 \).

### Final Answer
The two numbers are 6 and 4.

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