The LCM of two positive integers is twice the larger numbers . The difference of the smaller number and the GCD of the two numbers is 4. The smaller numbers is .

Options

A. 12
B. 6
C. 8
D. 10

Answer

To solve the problem step by step, we will use the information given in the question about the relationship between the two positive integers, their LCM, GCD, and the conditions provided.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the smaller number be \( x \) and the larger number be \( y \).

2. **Set Up the LCM Relationship:**
According to the question, the LCM of the two numbers is twice the larger number. Therefore, we can write:
\[ \text{LCM}(x, y) = 2y \]

3. **Set Up the GCD Relationship:**
The problem states that the difference between the smaller number and the GCD (Greatest Common Divisor) of the two numbers is 4. Let the GCD be \( \text{GCD}(x, y) = \text{HCF} \). Thus, we can write:
\[ x - \text{HCF} = 4 \quad \Rightarrow \quad x = \text{HCF} + 4 \]

4. **Use the Relationship Between LCM, GCD, and Product of Numbers:**
We know that:
\[ \text{LCM}(x, y) \times \text{HCF} = x \times y \]
Substituting the LCM from step 2, we have:
\[ 2y \times \text{HCF} = x \times y \]

5. **Substitute \( x \) from Step 3:**
Replace \( x \) in the equation:
\[ 2y \times \text{HCF} = (\text{HCF} + 4) \times y \]

6. **Rearrange the Equation:**
Expanding the right side gives:
\[ 2y \times \text{HCF} = \text{HCF} \times y + 4y \]
Rearranging the terms leads to:
\[ 2y \times \text{HCF} - \text{HCF} \times y = 4y \]
Factoring out \( \text{HCF} \) from the left side:
\[ \text{HCF} \times (2y - y) = 4y \]
Simplifying gives:
\[ \text{HCF} \times y = 4y \]

7. **Solve for HCF:**
Dividing both sides by \( y \) (assuming \( y \neq 0 \)):
\[ \text{HCF} = 4 \]

8. **Find the Smaller Number \( x \):**
Now substitute \( \text{HCF} \) back into the equation for \( x \):
\[ x = \text{HCF} + 4 = 4 + 4 = 8 \]

### Final Answer:
The smaller number \( x \) is \( 8 \).

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