Difficulty: Medium
Correct Answer: 39999999964
Explanation:
Introduction / Context:
This problem checks whether you can evaluate a large multiplication efficiently using identities instead of direct long multiplication. The numbers 199,994 and 200,006 are equally spaced around 200,000: one is 6 less and the other is 6 more. That structure is ideal for the identity (a - b)(a + b) = a^2 - b^2, also known as the difference of squares. Using it converts the multiplication into a simple subtraction of a small square from a large square, which is fast and less error-prone.
Given Data / Assumptions:
Concept / Approach:
Let a = 200,000 and b = 6. Then:
(200,000 - 6)(200,000 + 6) = 200,000^2 - 6^2.
Compute 200,000^2 accurately and subtract 36. This avoids multiplying six-digit numbers directly.
Step-by-Step Solution:
1) Rewrite the numbers around 200,000:
199,994 = 200,000 - 6
200,006 = 200,000 + 6
2) Apply difference of squares:
(200,000 - 6)(200,000 + 6) = 200,000^2 - 6^2
3) Compute the squares:
200,000^2 = (2 * 10^5)^2 = 4 * 10^10 = 40,000,000,000
6^2 = 36
4) Subtract:
40,000,000,000 - 36 = 39,999,999,964
Verification / Alternative check:
Estimate check: both numbers are about 200,000, so the product should be about 200,000 * 200,000 = 40,000,000,000. Because one is slightly smaller and the other slightly larger by the same amount, the product should be slightly less than 40,000,000,000 by 6^2 = 36. That matches exactly: 39,999,999,964.
Why Other Options Are Wrong:
• Values ending with 864 or 954 often come from subtracting 136 or 46 instead of 36, or placing digits incorrectly.
• 39,999,799,964 (or similar) indicates an incorrect square of 200,000 or incorrect subtraction position.
• 40,000,000,036 incorrectly adds 36 instead of subtracting it.
Common Pitfalls:
• Using (a - b)(a + b) = a^2 + b^2 (wrong sign).
• Squaring 200,000 incorrectly (missing zeros).
• Subtracting 36 from the wrong place causing digit shifts.
Final Answer:
39999999964
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