HCF is always

Options

A. multiple of LCM
B. Factor of LCM
C. divisible by LCM
D. a and c both

Answer

To answer the question "HCF is always...", we need to understand the relationship between the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of two numbers.

### Step-by-Step Solution:

1. **Define HCF and LCM**:
- HCF (Highest Common Factor) is the largest number that divides two or more numbers without leaving a remainder.
- LCM (Lowest Common Multiple) is the smallest number that is a multiple of two or more numbers.

2. **Choose Two Numbers**:
- Let's take two numbers, for example, \( a = 10 \) and \( b = 15 \).

3. **Find the Factors**:
- The factors of \( 10 \) are \( 1, 2, 5, 10 \).
- The factors of \( 15 \) are \( 1, 3, 5, 15 \).

4. **Identify the HCF**:
- The common factors of \( 10 \) and \( 15 \) are \( 1 \) and \( 5 \).
- Therefore, the HCF of \( 10 \) and \( 15 \) is \( 5 \).

5. **Find the LCM**:
- The multiples of \( 10 \) are \( 10, 20, 30, 40, \ldots \).
- The multiples of \( 15 \) are \( 15, 30, 45, 60, \ldots \).
- The smallest common multiple is \( 30 \), so the LCM of \( 10 \) and \( 15 \) is \( 30 \).

6. **Check the Relationship**:
- We need to check if the HCF is a factor of the LCM.
- Since \( 30 \div 5 = 6 \), we see that \( 5 \) is indeed a factor of \( 30 \).

7. **Conclusion**:
- Therefore, we conclude that the HCF is always a factor of the LCM.

### Final Answer:
HCF is always a factor of LCM.