If the HCF of a and b are 12 , where a,b are positive integers and` a gt b gt 12`,then what will be the values of a and b?

Options

A. a)12,24
B. b)24,12
C. c)24,36
D. d)36,24

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given that their highest common factor (HCF) is 12, and both \( a \) and \( b \) are positive integers with the condition \( a > b > 12 \).

### Step-by-Step Solution:

1. **Understanding HCF**: Since the HCF of \( a \) and \( b \) is 12, we can express \( a \) and \( b \) in terms of 12. We can write:
\[ a = 12m \quad \text{and} \quad b = 12n \]
where \( m \) and \( n \) are integers.

2. **Condition on \( m \) and \( n \)**: Given that \( a > b > 12 \), we can derive the following inequalities:
- From \( b > 12 \):
\[ 12n > 12 \implies n > 1 \]
- From \( a > b \):
\[ 12m > 12n \implies m > n \]

3. **Choosing Values for \( n \)**: Since \( n \) must be greater than 1, the smallest integer value for \( n \) is 2. Let's start with \( n = 2 \):
\[ n = 2 \implies b = 12 \times 2 = 24 \]
Since \( m \) must be greater than \( n \), the smallest integer value for \( m \) is 3:
\[ m = 3 \implies a = 12 \times 3 = 36 \]

4. **Verifying the Conditions**: Now we check the values:
- \( a = 36 \)
- \( b = 24 \)
- Check \( a > b > 12 \):
\[ 36 > 24 > 12 \quad \text{(True)} \]
- Check HCF:
\[ \text{HCF}(36, 24) = 12 \quad \text{(True)} \]

5. **Finding More Values**: We can also try other values for \( n \):
- If \( n = 3 \):
\[ b = 12 \times 3 = 36 \quad \text{(not valid since \( a \) must be greater than \( b \))} \]
- If \( n = 4 \):
\[ b = 12 \times 4 = 48 \]
Then \( m \) must be greater than 4, so \( m = 5 \):
\[ a = 12 \times 5 = 60 \]
- Check:
- \( a = 60 \)
- \( b = 48 \)
- \( 60 > 48 > 12 \quad \text{(True)} \)
- \( \text{HCF}(60, 48) = 12 \quad \text{(True)} \)

### Conclusion:
The values of \( a \) and \( b \) can be:
1. \( a = 36, b = 24 \)
2. \( a = 60, b = 48 \)