Calculate the value of the expression (61681 × 61681 − 31681 × 31681) ÷ 30000 by simplifying carefully, preferably using the difference of squares identity.

Introduction / Context:
This question involves large numbers, but it is structured to reward recognition of algebraic identities rather than raw computation. The numerator is a difference of two squares, which suggests the identity a² − b² = (a − b)(a + b). After applying this identity, we divide by 30000 to obtain the final value. Such simplifications are very common in aptitude exams.

Given Data / Assumptions:

• The expression to evaluate is (61681 × 61681 − 31681 × 31681) ÷ 30000.
• The numerator can be rewritten as 61681² − 31681².
• We can use the difference of squares identity a² − b² = (a − b)(a + b).
• All values are integers.

Concept / Approach:
Instead of squaring 61681 and 31681 directly, we apply the identity a² − b² = (a − b)(a + b). Once we factor the numerator, we often see a cancellation with the denominator. This drastically reduces computation and avoids mistakes with large products. After cancellation, we carry out the remaining multiplication to reach the final answer.

Step-by-Step Solution:
Rewrite the numerator as 61681² − 31681².Apply the difference of squares identity: a² − b² = (a − b)(a + b).Here, a = 61681 and b = 31681.Compute a − b: 61681 − 31681 = 30000.Compute a + b: 61681 + 31681 = 93362.So the numerator becomes 30000 × 93362.Now divide by 30000: (30000 × 93362) ÷ 30000 = 93362.

Verification / Alternative check:
The factor 30000 in the numerator cancels exactly with the denominator, which is clearly visible after using the difference of squares. If we attempted direct squaring, numbers would be extremely large, increasing the chance of error. The clean cancellation is a strong indication that 93362 is the correct result. Any other value would require incomplete or incorrect cancellation.

Why Other Options Are Wrong:
Values like 93352, 94362, 95362, and 92362 differ by small or moderate amounts from 93362. These values might result from small arithmetic mistakes such as incorrect addition when computing a + b or miscalculating a − b. However, when the identity and cancellation are applied accurately, 93362 is the only correct answer.

Common Pitfalls:
A common mistake is to try to compute 61681² and 31681² directly, which is time consuming and prone to digit errors. Another issue is misapplying the identity, for example writing (a − b)² instead of (a − b)(a + b). Some students also forget to divide by 30000 at the end. By writing each step clearly and cancelling the common factor, the solution becomes straightforward.

Final Answer:
The value of the given expression is 93362.