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

Options

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

Answer

To solve the problem step by step, we will denote the two positive integers as \( X \) (the larger number) and \( Y \) (the smaller number).

### Step 1: Set up the equations based on the problem statement.
We know from the problem that:
1. The LCM of the two numbers is twice the larger number:
\[ \text{LCM}(X, Y) = 2X \]
2. The difference between the smaller number and the GCD of the two numbers is 4:
\[ Y - \text{GCD}(X, Y) = 4 \]

### Step 2: Use the relationship between LCM, GCD, and the product of the numbers.
We can use the formula:
\[ \text{LCM}(X, Y) \times \text{GCD}(X, Y) = X \times Y \]
Substituting the LCM from step 1 into this equation gives:
\[ (2X) \times \text{GCD}(X, Y) = X \times Y \]

### Step 3: Express GCD in terms of Y.
From step 2, we can express GCD as:
\[ \text{GCD}(X, Y) = \frac{X \times Y}{2X} = \frac{Y}{2} \]
Now, substituting this GCD into the equation from step 1:
\[ Y - \frac{Y}{2} = 4 \]

### Step 4: Simplify the equation.
This simplifies to:
\[ \frac{Y}{2} = 4 \]

### Step 5: Solve for Y.
Multiplying both sides by 2 gives:
\[ Y = 8 \]

### Conclusion
The smaller number \( Y \) is 8.