What is the rule for 3, and why does adding the digits work?

A number is divisible by 3 if its digit sum is — because every power of ten leaves remainder 1 when divided by 3.

Since 10 ≡ 1 (mod 3), each digit contributes just itself to the remainder, so the whole number ≡ its digit sum (mod 3):

• 405 → 4 + 0 + 5 = 9 → divisible by 3. ✓
• 1,238 → 1 + 2 + 3 + 8 = 14 → not divisible by 3.

You can keep summing (14 → 5) since that preserves the remainder — the digital root.