If a - b = 4 and $a^2 + b^2$ = 40, where a and b are positive integers, then $a^3 + b^6$ is equal to :

Question

Accepted Answer

To solve the problem and justify why Option 2 is correct, we need to find the values of $a$ and $b$ that satisfy the given conditions and then calculate $a^3 + b^6$.

Given:
1. $a - b = 4$
2. $a^2 + b^2 = 40$

We need to find $a$ and $b$ such that both conditions are satisfied.

### Step 1: Solve the System of Equations

From the first equation, express $a$ in terms of $b$:
$$ a = b + 4 $$

Substitute $a = b + 4$ into the second equation:
$$ (b + 4)^2 + b^2 = 40 $$

Expand and simplify:
$$ b^2 + 8b + 16 + b^2 = 40 $$
$$ 2b^2 + 8b + 16 = 40 $$
$$ 2b^2 + 8b - 24 = 0 $$

Divide the entire equation by 2:
$$ b^2 + 4b - 12 = 0 $$

Factor the quadratic equation:
$$ (b + 6)(b - 2) = 0 $$

Thus, $b = -6$ or $b = 2$. Since $b$ is a positive integer, we take $b = 2$.

Substitute $b = 2$ back into $a = b + 4$:
$$ a = 2 + 4 = 6 $$

### Step 2: Calculate $a^3 + b^6$

Now that we have $a = 6$ and $b = 2$, calculate $a^3 + b^6$:
$$ a^3 = 6^3 = 216 $$
$$ b^6 = 2^6 = 64 $$

Therefore:
$$ a^3 + b^6 = 216 + 64 = 280 $$

### Conclusion

The calculation shows that $a^3 + b^6 = 280$, which corresponds to Option 2.

### Explanation of Incorrect Options

1. **Option 1 (264):** This value does not match the calculated result of 280.
2. **Option 3 (300):** This value exceeds the calculated result of 280.
3. **Option 4 (324):** This value also exceeds the calculated result of 280.

Thus, Option 2 is the correct answer, as it accurately reflects the value of $a^3 + b^6$ given the conditions of the problem.