If `a - b = 1 and a ^(3) - b ^(3) = 61,` then the value of ab will be ?

Options

A. `-20`
B. 20
C. 30
D. 60

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations**:
\[ a - b = 1 \]
\[ a^3 - b^3 = 61 \]

2. **Using the Identity for Difference of Cubes**:
We can use the identity for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Substituting the value of \(a - b\):
\[ a^3 - b^3 = 1 \cdot (a^2 + ab + b^2) = a^2 + ab + b^2 \]

3. **Setting Up the Equation**:
Now, we can set up the equation:
\[ a^2 + ab + b^2 = 61 \]

4. **Expressing \(a^2 + b^2\)**:
We know that:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
Substituting \(a - b = 1\):
\[ a^2 + b^2 = 1^2 + 2ab = 1 + 2ab \]

5. **Substituting Back**:
Now substitute \(a^2 + b^2\) back into the equation:
\[ 1 + 2ab + ab = 61 \]
This simplifies to:
\[ 1 + 3ab = 61 \]

6. **Solving for \(ab\)**:
Now, isolate \(ab\):
\[ 3ab = 61 - 1 \]
\[ 3ab = 60 \]
\[ ab = \frac{60}{3} = 20 \]

7. **Final Answer**:
Thus, the value of \(ab\) is:
\[ \boxed{20} \]