Number System — Compute the exact product: 1014 × 986. Use any reliable method (near-1000 identity or standard multiplication) and choose the correct value.

Introduction / Context:
This multiplication is tailor-made for the identity (1000 + a) × (1000 − a) = 1000^2 − a^2. Spotting symmetry around 1000 not only speeds up the computation but also reduces the chance of digit errors compared to full long multiplication.

Given Data / Assumptions:

• We must compute 1014 × 986 exactly.
• Notice 1014 = 1000 + 14 and 986 = 1000 − 14.
• Apply the difference of squares with a = 14.

Concept / Approach:
Use (1000 + 14) × (1000 − 14) = 1000^2 − 14^2. This uses fewer steps and produces an exact result quickly. Calculating 14^2 is simple and ensures accuracy in the final subtraction.

Step-by-Step Solution:
1) Recognize the pattern: (1000 + 14)(1000 − 14) = 1000000 − 14^2.2) Compute 14^2 = 196.3) Subtract: 1,000,000 − 196 = 999,804.4) Therefore, 1014 × 986 = 999,804.

Verification / Alternative check:
Rough magnitude check: 1014 × 986 is slightly less than 1014 × 1000 (≈ 1,014,000) and slightly less than 1000 × 1000. The exact identity confirms the precise value 999,804.

Why Other Options Are Wrong:
998904, 998814, 998804, and 998744 all subtract too much from 1,000,000 (larger than 196), reflecting errors in computing 14^2 or in the final subtraction.

Common Pitfalls:
Using 14^2 = 194 or 200 by mistake; dropping zeros when handling 1,000,000; confusing (a + b)(a − b) with a^2 − b^2 and reversing terms.

Final Answer:
999804