The HCF of the two numbers a and b is 12. Which of the following can be the LCM of a and b ?

Options

A. 44
B. 65
C. 76
D. 96

Explanation

To determine which of the given options can be the LCM of two numbers \( a \) and \( b \) when their HCF is 12, we can follow these steps:

### Step 1: Understand the relationship between HCF and LCM
The relationship between the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two numbers \( a \) and \( b \) is given by the formula:
\[ \text{HCF} \times \text{LCM} = a \times b \]
Since we know the HCF is 12, we can express this as:
\[ 12 \times \text{LCM} = a \times b \]
This means that the LCM must be a multiple of the HCF.

### Step 2: Identify the possible LCM values
To find possible LCM values, we need to check if the given options are multiples of 12.

### Step 3: Check each option
Let's check the options one by one:

1. **Option 1: 44**
- Check if 44 is divisible by 12:
\[ 44 \div 12 = 3.67 \quad (\text{not an integer}) \]
- **Conclusion**: 44 is not a multiple of 12.

2. **Option 2: 65**
- Check if 65 is divisible by 12:
\[ 65 \div 12 = 5.42 \quad (\text{not an integer}) \]
- **Conclusion**: 65 is not a multiple of 12.

3. **Option 3: 76**
- Check if 76 is divisible by 12:
\[ 76 \div 12 = 6.33 \quad (\text{not an integer}) \]
- **Conclusion**: 76 is not a multiple of 12.

4. **Option 4: 96**
- Check if 96 is divisible by 12:
\[ 96 \div 12 = 8 \quad (\text{is an integer}) \]
- **Conclusion**: 96 is a multiple of 12.

### Step 4: Final conclusion
From the checks above, the only option that can be the LCM of \( a \) and \( b \) when their HCF is 12 is **96**.