$217 \times 217 + 183 \times 183 = x$

### Concept & Formula Just like the previous problem, we are evaluating $a^2 + b^2$. Here, $217$ and $183$ are precisely equidistant from $200$. By exploiting this symmetry, we can collapse the expression using a standard algebraic formula. $$(x + y)^2 + (x - y)^2 = 2(x^2 + y^2)$$ ### Step-by-Step Solution * Define the common base and the offset: The base $x = 200$, and the offset $y = 17$. $$217 = 200 + 17$$ $$183 = 200 - 17$$ * Substitute these into the combined identity: $$(200 + 17)^2 + (200 - 17)^2 = 2 \times ((200)^2 + (17)^2)$$ * Evaluate the squared terms inside the parenthesis: $$(200)^2 = 40000$$ $$(17)^2 = 289$$ * Add and multiply by 2: $$2 \times (40000 + 289) = 2 \times 40289 = 80578$$ ### Exam Strategy & Shortcut Use the "last two digits" shortcut derived from the formula $2(x^2 + y^2)$. Since $x=200$, $x^2$ ends in $0000$. The last digits of the entire sum will entirely depend on $2 \times y^2$. $$2 \times (17)^2 = 2 \times 289 = 578$$ The answer must end in exactly $578$. Scanning the options, only option (b) $80578$ satisfies this rule! You can circle the answer without calculating the front digits. ### Common Pitfall Getting intimidated by the numbers and guessing, or making a calculation error while attempting to find $217 \times 217$ manually. Always look for a clean base (like 100, 200, 50) hiding between the given numbers. ### Final Answer **Therefore, the correct answer is 80578.**