Using the algebraic identity (a - b)(a + b) = a^2 - b^2, compute the product 9997 × 10003 by treating the numbers as 10000 - 3 and 10000 + 3, and then choose the correct result.

Introduction / Context:
This simplification question is designed to test your familiarity with special algebraic products, particularly the identity (a - b)(a + b) = a^2 - b^2. Instead of multiplying 9997 and 10003 directly, which is time consuming and error prone, you can exploit the fact that both numbers lie close to 10000 and are symmetrical around it. This is a classic mental math trick used in many quantitative aptitude exams.

Given Data / Assumptions:

• We need to compute 9997 × 10003.
• Recognise that 9997 = 10000 - 3.
• Recognise that 10003 = 10000 + 3.
• The identity (a - b)(a + b) = a^2 - b^2 is valid for all real numbers a and b.
• No calculator is required if the identity is applied correctly.

Concept / Approach:
By setting a = 10000 and b = 3, the product 9997 × 10003 becomes (a - b)(a + b). According to the identity, this equals a^2 - b^2. Calculating a^2 is straightforward for powers of 10, and b^2 is a small integer. Subtracting b^2 from a^2 gives the final result in a few seconds, making this method both quick and reliable.

Step-by-Step Solution:
Write 9997 as 10000 - 3 and 10003 as 10000 + 3. Recognise the pattern (a - b)(a + b) with a = 10000 and b = 3. Apply the identity: (a - b)(a + b) = a^2 - b^2. Compute a^2 = 10000^2 = 100000000 (eight zeros after 1). Compute b^2 = 3^2 = 9. Subtract: 100000000 - 9 = 99999991.
Verification / Alternative check:
You can perform a rough check by approximating the numbers. Both 9997 and 10003 are close to 10000, so their product should be very close to 100000000. Since 9997 is slightly less than 10000 and 10003 is slightly more, the product will be slightly less than 100000000, which matches 99999991. You could also perform traditional multiplication digit by digit to confirm, but this is more tedious and prone to manual mistakes.

Why Other Options Are Wrong:
Options b, c, d, and e are numbers that look visually similar, but they correspond to subtracting 89, 1009, 889, or another incorrect quantity from 100000000 rather than 9. They may arise from miscalculating b^2 or from misplacing zeros in a^2. Only 99999991 fits the structure of 100000000 - 9 and respects the pattern from the identity.

Common Pitfalls:
A common error is to wrongly take a as 9997 and b as 3, leading to an incorrect (a - b)(a + b) representation. Another mistake is miscomputing 10000^2 by writing 10000000 instead of 100000000, which shifts all subsequent digits. Remember that 10^4 squared is 10^8, so there must be eight zeros. If you handle these details carefully, the identity method is very reliable.

Final Answer:
Using the identity (a - b)(a + b) = a^2 - b^2, the product 9997 × 10003 is 99999991.