Simplify the expression 2003 × 2004 − 2001 × 2002 by using algebraic identities or efficient calculation techniques, and find its exact numerical value.

Introduction / Context:
This question tests recognition of algebraic patterns and efficient computation with large numbers. Instead of multiplying each pair and then subtracting, you can use algebraic identities involving differences of products. Recognising such structures can save time in competitive examinations where direct multiplication is slow and error prone.

Given Data / Assumptions:

• The expression to evaluate is 2003 × 2004 − 2001 × 2002.
• All numbers are integers around 2000.
• We look for an algebraic identity to simplify the difference of two products.

Concept / Approach:
Observe that 2003 and 2004 are consecutive integers, as are 2001 and 2002. We can write the expression in a more general form as (n + 2)(n + 3) − n(n + 1) for n = 2001. Then we expand each product and simplify. The result will be a linear expression in n which can easily be evaluated. This method is much faster than carrying out every multiplication in full.

Step-by-Step Solution:
Let n = 2001.Then 2003 = n + 2 and 2004 = n + 3, and 2001 = n, 2002 = n + 1.Rewrite the expression: 2003 × 2004 − 2001 × 2002 = (n + 2)(n + 3) − n(n + 1).Expand (n + 2)(n + 3) = n^2 + 5n + 6.Expand n(n + 1) = n^2 + n.Now subtract: (n^2 + 5n + 6) − (n^2 + n) = n^2 + 5n + 6 − n^2 − n.Simplify: 4n + 6.Substitute n = 2001: 4 × 2001 + 6 = 8004 + 6 = 8010.

Verification / Alternative check:
As a direct check, approximate the products. 2003 × 2004 is slightly more than 2000 × 2000, or about 4,012,000, while 2001 × 2002 is also close to 4,008,000. The difference should be a few thousand, and 8010 fits that expectation. A calculator or careful long multiplication would confirm that both products differ by exactly 8010, verifying the algebraic method.

Why Other Options Are Wrong:
Values like 8020, 8030, and 8040 come from small arithmetic mistakes, such as adding or subtracting an extra 10 or 20 when combining terms. 8000 is a rough estimate that ignores the exact expansion. Only 8010 matches the simplified expression 4n + 6 when n = 2001.

Common Pitfalls:
Students often attempt full multiplication with large numbers and introduce digit errors. Others may not notice the convenient pattern of consecutive integers and miss the chance to simplify using algebra. Recognising expressions of the form (n + a)(n + b) − n(n + c) and expanding them symbolically can greatly speed up such calculations in exams.

Final Answer:
The value of 2003 × 2004 − 2001 × 2002 is 8010.