If $a$, $b$, $c$ are integers; $a^2 + b^2 = 45$ and $b^2 + c^2 = 40$, then the values of $a$, $b$ and $c$ respectively are

### Concept & Strategy This problem involves solving a system of nonlinear Diophantine equations (equations where the solutions must be integers). We can use algebraic manipulation or simply check the given options to find a valid integer set. ### Step-by-Step Solution Given equations: 1) $a^2 + b^2 = 45$ 2) $b^2 + c^2 = 40$ Subtract equation (2) from equation (1): $(a^2 + b^2) - (b^2 + c^2) = 45 - 40$ $a^2 - c^2 = 5$ Using the difference of squares formula: $(a - c)(a + c) = 5$ Since $a$ and $c$ are integers, their sum and difference must also be integers. The only integer factors of 5 are $(1, 5)$ and $(-1, -5)$. Assuming positive integers for a moment: $a - c = 1$ $a + c = 5$ Adding both equations: $2a = 6 \implies a = 3$. Subtracting: $2c = 4 \implies c = 2$. Now, substitute $a = 3$ back into equation (1) to find $b$: $3^2 + b^2 = 45$ $9 + b^2 = 45$ $b^2 = 36 \implies b = 6$ (or $-6$). So the valid positive integer triplet is $a = 3, b = 6, c = 2$. ### Exam Strategy & Shortcut **Option Verification:** Instead of solving mathematically, directly plug the options into the original equations. Check Option (a): $a=2, b=6, c=3$ $a^2 + b^2 = 4 + 36 = 40$ (But it must be 45). Incorrect. Check Option (b): $a=3, b=2, c=6$ $a^2 + b^2 = 9 + 4 = 13$ (But it must be 45). Incorrect. Check Option (c): $a=5, b=4, c=3$ $a^2 + b^2 = 25 + 16 = 41$ (But it must be 45). Incorrect. Since none of the options perfectly match the valid set $(3, 6, 2)$, the answer is clearly "None of these". ### Common Pitfall Students might calculate the correct values $(a=3, b=6, c=2)$ and hastily select an option that contains these numbers in the wrong order, such as picking (b) just because it has a 3, a 2, and a 6. Pay strict attention to the word "respectively". ### Final Answer Therefore, the correct answer is **None of these**.