If a and b are two positive integers, then the relation between their L.C.M and H.C.F will be :

Options

A. L.C.M. gt H. C. F.
B. H.C.F gt L. C. M
C. H.C.F = L. C. M
D. None of these

Answer

To find the relation between the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two positive integers \( a \) and \( b \), we can use the following steps:

### Step-by-Step Solution:

1. **Understanding LCM and HCF**:
- The LCM of two numbers is the smallest number that is a multiple of both.
- The HCF of two numbers is the largest number that divides both.

2. **Using the relationship**:
- There is a well-known relationship between LCM and HCF of two numbers \( a \) and \( b \):
\[ \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b \]

3. **Assuming values for \( a \) and \( b \)**:
- Let's assume \( a = 1 \) and \( b = 2 \) (as per the example in the transcript).
- Calculate \( \text{HCF}(1, 2) \):
- The only divisor of both 1 and 2 is 1.
- So, \( \text{HCF}(1, 2) = 1 \).

4. **Calculate LCM(1, 2)**:
- The multiples of 1 are: 1, 2, 3, 4, ...
- The multiples of 2 are: 2, 4, 6, 8, ...
- The smallest common multiple is 2.
- So, \( \text{LCM}(1, 2) = 2 \).

5. **Verifying the relationship**:
- Now, using the relationship:
\[ \text{LCM}(1, 2) \times \text{HCF}(1, 2) = 2 \times 1 = 2 \]
- Also, \( 1 \times 2 = 2 \).
- Both sides are equal, confirming our relationship.

6. **Conclusion**:
- From the calculations, we see that:
\[ \text{LCM}(a, b) = 2 \quad \text{and} \quad \text{HCF}(a, b) = 1 \]
- Therefore, we can conclude that for any two positive integers \( a \) and \( b \):
\[ \text{LCM}(a, b) \geq \text{HCF}(a, b) \]
- In general, the LCM is always greater than or equal to the HCF.