Quickly calculate the product 9997 × 10003 by using the algebraic identity (a − b)(a + b) = a^2 − b^2 instead of long multiplication.

Introduction / Context:

This arithmetic problem is designed to reward the use of algebraic identities instead of brute force multiplication. The numbers 9997 and 10003 are very close to 10000, so the product can be evaluated efficiently using the difference of squares formula. Recognising such patterns is a valuable shortcut in mental math and competitive exams.

Given Data / Assumptions:

• We want to compute 9997 × 10003.
• Both numbers are near 10000.
• We can exploit the identity (a − b)(a + b) = a^2 − b^2.
• No calculators are required; the problem is structured for exact mental or written computation.

Concept / Approach:

Write both numbers in the form 10000 ± 3. Then the product becomes (10000 − 3)(10000 + 3), which fits the identity (a − b)(a + b) = a^2 − b^2. This reduces the problem to squaring 10000 and subtracting 3^2. Squaring 10000 is straightforward, and subtracting 9 gives the final answer with minimal effort.

Step-by-Step Solution:

Verification / Alternative check:

You can perform a rough check by estimation. Since both numbers are very close to 10000, the product should be close to 10000 × 10000 = 100000000. The answer 99999991 is only 9 less than this, which matches the identity based calculation. This confirms that the algebraic approach is consistent.

Why Other Options Are Wrong:

The options 9999991, 99999911, 9999911, and 99990003 differ significantly from 100000000 by larger amounts or in the wrong pattern. None of them equals 100000000 − 9. Only 99999991 matches the exact difference of squares calculation.

Common Pitfalls:

A common mistake is to overlook the identity and attempt full long multiplication, which is slow and more error prone. Others may miscompute 10000^2, forgetting that it produces eight zeros, or may subtract 3 instead of 9. Remembering the identity (a − b)(a + b) = a^2 − b^2 and applying it carefully prevents these errors.

Final Answer:

The value of 9997 × 10003 is 99999991.