Is it possible to have two numbers whose HCF is 20 and LCM is 630? Give reason.

Answer

To determine if it is possible to have two numbers whose HCF (Highest Common Factor) is 20 and LCM (Least Common Multiple) is 630, we can use the relationship between HCF and LCM.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The relationship between HCF and LCM of two numbers \( a \) and \( b \) is given by the formula:
\[ \text{HCF} \times \text{LCM} = a \times b \]
This means that the product of the HCF and LCM of two numbers equals the product of the numbers themselves.

2. **Substituting the Given Values**:
We know that:
- HCF = 20
- LCM = 630

Substituting these values into the formula:
\[ 20 \times 630 = a \times b \]

3. **Calculating the Product**:
Now, we calculate the left-hand side:
\[ 20 \times 630 = 12600 \]
Therefore, we have:
\[ a \times b = 12600 \]

4. **Checking Divisibility**:
For the two numbers \( a \) and \( b \) to exist with the given HCF and LCM, the LCM must be divisible by the HCF. We need to check if:
\[ \frac{\text{LCM}}{\text{HCF}} = \frac{630}{20} \]

5. **Performing the Division**:
Now, we perform the division:
\[ \frac{630}{20} = 31.5 \]
Since the result is not an integer, it indicates that 630 is not completely divisible by 20.

6. **Conclusion**:
Since the LCM is not divisible by the HCF, it is not possible to have two numbers whose HCF is 20 and LCM is 630.

### Final Answer:
No, it is not possible to have two numbers whose HCF is 20 and LCM is 630 because the LCM must be divisible by the HCF, and in this case, it is not.