Greatest four-digit multiple: Find the greatest four-digit number that is divisible by each of 12, 18, 21, and 28.

Introduction / Context:
The largest four-digit number divisible by several integers must be a multiple of their least common multiple (LCM). Compute the LCM, then take the largest multiple not exceeding 9999.

Given Data / Assumptions:

• Divisors: 12, 18, 21, 28
• Upper limit: 9999 (largest four-digit number)

Concept / Approach:
LCM via prime powers: choose the highest power of each prime appearing in the prime factorizations. Then compute floor(9999 / LCM) * LCM.

Step-by-Step Solution:

Verification / Alternative check:
Check divisibility: 9828 / 12 = 819, /18 = 546, /21 = 468, /28 = 351 — all integers.

Why Other Options Are Wrong:
9864 and 9848 are not multiples of 252; 9636 is a multiple of 252 but smaller than 9828; 9996 is not divisible by all listed numbers.

Common Pitfalls:
Computing LCM incorrectly (e.g., missing 3^2) or multiplying by 40 (which gives 10080, exceeding four digits).

Final Answer:
9828