Using the identity for the difference of squares, calculate the value of the numerical expression (61681^2 − 31681^2) ÷ 30000.

Introduction / Context:
This question demonstrates how algebraic identities can simplify apparently huge numerical calculations. Directly squaring 61681 and 31681 would be very tedious. However, the expression has the special form a^2 − b^2, which can be rewritten as (a − b)(a + b). Recognizing and using this identity simplifies the calculation dramatically.

Given Data / Assumptions:

• The expression to evaluate is (61681^2 − 31681^2) ÷ 30000.
• We assume normal integer arithmetic.
• The numbers 61681 and 31681 are just large integers with no further hidden structure.
• The task is to compute the exact integer result.

Concept / Approach:
Use the identity a^2 − b^2 = (a − b)(a + b). Here a = 61681 and b = 31681. After factorization, we see that the numerator becomes (a − b)(a + b), and in this specific problem a − b equals 30000, which matches the divisor in the denominator. This allows immediate cancellation and leaves only the sum a + b to compute, which is manageable even by mental arithmetic.

Step-by-Step Solution:
Let a = 61681 and b = 31681.Compute a − b: 61681 − 31681 = 30000.Compute a + b: 61681 + 31681 = 93362.Using the identity a^2 − b^2 = (a − b)(a + b), the numerator becomes 30000 * 93362.The full expression is (30000 * 93362) ÷ 30000.Cancel the factor 30000 in numerator and denominator: the result is simply 93362.

Verification / Alternative check:
An approximate check can be done by noticing that 61681 is roughly 62000 and 31681 is roughly 32000. The difference of squares of those rough values, divided by 30000, would be close to the sum of the two numbers, which is about 94000. Our exact answer 93362 is near that rough estimate, indicating that the calculation is plausible. More precise checks would involve computing the exact product, but the algebraic identity is exact and guarantees the correctness once the basic subtraction and addition are correct.

Why Other Options Are Wrong:

• Values like 94362, 95362, and 96362 are close but do not equal the exact sum 61681 + 31681.
• 93352 differs by 10 from the correct answer and would result from a minor arithmetic mistake when adding 61681 and 31681.
• Only 93362 corresponds exactly to a + b after correct computation.

Common Pitfalls:

• Failing to recognize the difference of squares pattern and trying to square the large numbers directly.
• Subtracting or adding the large numbers incorrectly, especially when carrying digits.
• Forgetting to cancel the common factor 30000 in numerator and denominator after factorization.

Final Answer:
93362