Find the greatest four digit number which is divisible by 3, 6, and 9.

Concept: To find the greatest number (in a given range) that is divisible by several numbers together, first find the LCM of those numbers -- every multiple of the LCM is automatically divisible by each of them individually.

1. Find the LCM of 3, 6, and 9. Since 6 = 2×3 and 9 = 3×3, the LCM is 2×3×3 = 18.

2. The greatest four-digit number is 9999.

3. Divide 9999 by 18: 9999 = 18 × 555 + 9, so the remainder is 9.

4. Subtract the remainder from 9999 to get the largest multiple of 18 not exceeding it: 9999 − 9 = 9990.

Cross-check: the digit sum of 9990 is 9+9+9+0 = 27, which is divisible by both 3 and 9, and 9990 is even, so it is divisible by 6 as well -- confirming 9990 satisfies all three conditions and is exactly 18 × 555.