The LCM of two positive integers is thrice the larger number. The difference of the smaller number and the GCD of the two numbers is 8. The smaller number is :

Options

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

Answer

To solve the problem, we will follow these steps:

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

2. **Set Up the Equations**:
- According to the problem, the LCM of the two numbers is thrice the larger number:
\[ \text{LCM}(L, S) = 3L \]
- The difference between the smaller number and the GCD (let's denote GCD as \( G \)) is 8:
\[ S - G = 8 \quad \Rightarrow \quad G = S - 8 \]

3. **Use the Relationship Between LCM, GCD, and the Numbers**:
- The relationship between LCM, GCD, and the two numbers is given by:
\[ \text{LCM}(L, S) \cdot G = L \cdot S \]
- Substituting the expressions we have:
\[ (3L)(G) = L \cdot S \]
- Substitute \( G \) with \( S - 8 \):
\[ 3L(S - 8) = L \cdot S \]

4. **Simplify the Equation**:
- Divide both sides by \( L \) (assuming \( L \neq 0 \)):
\[ 3(S - 8) = S \]
- Distributing the 3:
\[ 3S - 24 = S \]

5. **Rearrange the Equation**:
- Move \( S \) to the left side:
\[ 3S - S = 24 \]
- This simplifies to:
\[ 2S = 24 \]

6. **Solve for \( S \)**:
- Divide both sides by 2:
\[ S = \frac{24}{2} = 12 \]

7. **Conclusion**:
- The smaller number \( S \) is 12.

### Final Answer:
The smaller number is **12**.