What is the rule for 9, and how is it different from the rule for 3?

Same idea — digit sum divisible by 9 → the number is — but ÷3 does not imply ÷9.

Since 10 ≡ 1 (mod 9) too, the number ≡ its digit sum (mod 9):

• 2,880 → 2 + 8 + 8 + 0 = 18 → 1 + 8 = 9 → divisible by 9 (and 3). ✓
• best trap: 12 → digit sum 3 → divisible by 3 but not 9.

So a digit sum divisible by 3 guarantees ÷3; you additionally need it divisible by 9 for ÷9.