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 step by step, we will follow the mathematical reasoning outlined in the video transcript.

### Step 1: Define the Variables
Let the two numbers be \( X \) and \( Y \).

### Step 2: Set Up the Equations
According to the problem:
1. The product of the LCM and HCF of the two numbers is 24:
\[ \text{LCM}(X, Y) \times \text{HCF}(X, Y) = X \times Y = 24 \]
2. The difference between the two numbers is 2:
\[ X - Y = 2 \]

### Step 3: Express One Variable in Terms of the Other
From the second equation, we can express \( X \) in terms of \( Y \):
\[ X = Y + 2 \]

### Step 4: Substitute into the First Equation
Now, substitute \( X \) in the first equation:
\[ (Y + 2) \times Y = 24 \]
Expanding this gives:
\[ Y^2 + 2Y = 24 \]

### Step 5: Rearrange the Equation
Rearranging the equation, we get:
\[ Y^2 + 2Y - 24 = 0 \]

### Step 6: Factor the Quadratic Equation
Next, we will factor the quadratic equation. We need two numbers that multiply to -24 and add to 2. These numbers are 6 and -4:
\[ (Y + 6)(Y - 4) = 0 \]

### Step 7: Solve for \( Y \)
Setting each factor to zero gives us:
1. \( Y + 6 = 0 \) → \( Y = -6 \) (not valid since we are looking for positive numbers)
2. \( Y - 4 = 0 \) → \( Y = 4 \)

### Step 8: Find \( X \)
Now, substitute \( Y = 4 \) back into the equation for \( X \):
\[ X = Y + 2 = 4 + 2 = 6 \]

### Step 9: State the Final Answer
The two numbers are \( X = 6 \) and \( Y = 4 \).

### Summary of the Solution
The two numbers are 6 and 4.

---