The HCF (GCD) of a, b is 12, a, b are positive integers and `a gt b gt 12` . The smallest values of (a, b) are respectively

Options

A. 12,24
B. 24,12
C. 24,36
D. 36,24

Answer

To find the smallest values of \( a \) and \( b \) given that the HCF (GCD) of \( a \) and \( b \) is 12, and \( a > b > 12 \), we can follow these steps:

### Step 1: Understanding the HCF
Since the HCF of \( a \) and \( b \) is 12, we can express \( a \) and \( b \) in terms of their HCF:
\[ a = 12m \quad \text{and} \quad b = 12n \]
where \( m \) and \( n \) are coprime integers (i.e., their HCF is 1).

### Step 2: Setting up inequalities
From the problem statement, we know:
\[ a > b > 12 \]
Substituting the expressions for \( a \) and \( b \):
\[ 12m > 12n > 12 \]
Dividing the entire inequality by 12 gives:
\[ m > n > 1 \]

### Step 3: Finding the smallest values for \( m \) and \( n \)
Since \( m \) and \( n \) must be coprime integers greater than 1, the smallest possible values for \( n \) and \( m \) that satisfy \( m > n \) are:
- Let \( n = 2 \)
- Then \( m = 3 \) (since 3 is the next integer greater than 2)

### Step 4: Calculating \( a \) and \( b \)
Now substituting back to find \( a \) and \( b \):
\[ b = 12n = 12 \times 2 = 24 \]
\[ a = 12m = 12 \times 3 = 36 \]

### Conclusion
Thus, the smallest values of \( (a, b) \) are:
\[ (a, b) = (36, 24) \]