Can two numbers have 16 as their HCF and 204 as their LCM ? Give reason.

Answer

To determine if two numbers can have 16 as their HCF (Highest Common Factor) and 204 as their LCM (Least Common Multiple), we can follow these steps:

### Step-by-Step Solution:

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

2. **Relationship Between HCF and LCM**:
- A key property of HCF and LCM is that for any two numbers \(a\) and \(b\):
\[ \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \]
- This means that the product of the HCF and LCM of two numbers is equal to the product of the numbers themselves.

3. **Checking Divisibility**:
- If the HCF is 16, then the LCM (204) must be divisible by the HCF (16).
- We need to check if \(204\) is divisible by \(16\).

4. **Performing the Division**:
- Divide \(204\) by \(16\):
\[ 204 \div 16 = 12.75 \]
- This means that \(204\) is not an integer multiple of \(16\).

5. **Conclusion**:
- Since \(204\) is not divisible by \(16\), it is not possible for two numbers to have \(16\) as their HCF and \(204\) as their LCM.

### Final Answer:
No, two numbers cannot have 16 as their HCF and 204 as their LCM because 204 is not divisible by 16.

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