The HCF (GCD) of a, b is 12, a, b are positive integers and `a gt b gt 12`. The smallest value 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: Express \( a \) and \( b \) in terms of their HCF
Since the HCF of \( a \) and \( b \) is 12, we can express \( a \) and \( b \) as:
\[ a = 12x \quad \text{and} \quad b = 12y \]
where \( x \) and \( y \) are coprime integers (i.e., their HCF is 1).

### Step 2: Set up the inequalities
From the problem, we know:
\[ a > b > 12 \]
Substituting the expressions for \( a \) and \( b \):
\[ 12x > 12y > 12 \]
We can simplify this by dividing all parts of the inequality by 12:
\[ x > y > 1 \]

### Step 3: Find the smallest coprime integers \( x \) and \( y \)
To satisfy \( x > y > 1 \), we can start with the smallest values for \( y \):
- If \( y = 2 \), then the smallest integer \( x \) that is greater than 2 and coprime to 2 is \( x = 3 \).

### Step 4: Calculate \( a \) and \( b \)
Using \( x = 3 \) and \( y = 2 \):
\[ a = 12x = 12 \times 3 = 36 \]
\[ b = 12y = 12 \times 2 = 24 \]

### Step 5: Verify the conditions
We need to check:
- \( a > b \): \( 36 > 24 \) (True)
- \( b > 12 \): \( 24 > 12 \) (True)

Both conditions are satisfied.

### Conclusion
The smallest values of \( a \) and \( b \) are:
\[ a = 36, \quad b = 24 \]