The H.C.F. of two numbers is 8. Which one of the following can never be their L.C.M . ?

Options

A. 24
B. 48
C. 56
D. 60

Answer

To solve the problem, we need to determine which of the given options can never be the LCM of two numbers when their HCF is 8.

### Step-by-Step Solution:

1. **Understanding HCF and LCM**:
- The HCF (Highest Common Factor) of two numbers is the largest number that divides both of them. In this case, the HCF is 8.
- The LCM (Least Common Multiple) is the smallest number that is a multiple of both numbers.

2. **Identifying the Form of the Numbers**:
- Since the HCF is 8, both numbers must be multiples of 8. We can express the two numbers as:
- Number 1 = 8a
- Number 2 = 8b
- Here, 'a' and 'b' are integers that have no common factors other than 1 (since 8 is the common factor).

3. **Finding the LCM**:
- The formula for LCM in terms of HCF is:
\[ \text{LCM} = \frac{\text{Number 1} \times \text{Number 2}}{\text{HCF}} \]
- Substituting the expressions for Number 1 and Number 2:
\[ \text{LCM} = \frac{(8a) \times (8b)}{8} = 8ab \]
- This shows that the LCM must also be a multiple of 8.

4. **Analyzing the Options**:
- We need to check which of the given options is not a multiple of 8.
- Let's evaluate the options:
- **Option 1: 24** - \( 24 = 8 \times 3 \) (multiple of 8)
- **Option 2: 48** - \( 48 = 8 \times 6 \) (multiple of 8)
- **Option 3: 56** - \( 56 = 8 \times 7 \) (multiple of 8)
- **Option 4: 60** - \( 60 \div 8 = 7.5 \) (not a multiple of 8)

5. **Conclusion**:
- The only number that is not a multiple of 8 is 60. Therefore, 60 can never be the LCM of two numbers whose HCF is 8.

### Final Answer:
60 can never be their LCM.