If a and b be positive integers such that `a^(2) - b^(2)= 19` then the value of a is

Options

A. 19
B. 20
C. 9
D. 10

Answer

To solve the equation \( a^2 - b^2 = 19 \) where \( a \) and \( b \) are positive integers, we can use the identity for the difference of squares:

1. **Rewrite the equation using the difference of squares identity**:
\[ a^2 - b^2 = (a + b)(a - b) \]
Therefore, we can express the equation as:
\[ (a + b)(a - b) = 19 \]

2. **Identify the factors of 19**:
Since 19 is a prime number, its only positive factors are 1 and 19. Thus, we can set:
\[ a + b = 19 \quad \text{and} \quad a - b = 1 \]

3. **Set up the system of equations**:
We now have two equations:
\[ a + b = 19 \quad \text{(1)} \]
\[ a - b = 1 \quad \text{(2)} \]

4. **Add the two equations**:
Adding equations (1) and (2):
\[ (a + b) + (a - b) = 19 + 1 \]
This simplifies to:
\[ 2a = 20 \]

5. **Solve for \( a \)**:
Dividing both sides by 2:
\[ a = 10 \]

6. **Substitute back to find \( b \)**:
Now, substitute \( a = 10 \) back into equation (1):
\[ 10 + b = 19 \]
Solving for \( b \):
\[ b = 19 - 10 = 9 \]

7. **Conclusion**:
The values of \( a \) and \( b \) are \( a = 10 \) and \( b = 9 \). Therefore, the value of \( a \) is:
\[ \boxed{10} \]