What is the rule for 6?

Divisible by 6 exactly when it passes both the ÷2 and ÷3 tests.

6 = 2 × 3, and 2 and 3 share no factors, so a number divisible by both is divisible by their product:

• 138 → even ✓ and digit sum 1+3+8 = 12 (÷3) ✓ → divisible by 6. (138 ÷ 6 = 23.) ✓
• 114 → even ✓, digit sum 6 (÷3) ✓ → ÷6. But 116 → even but digit sum 8 (not ÷3) → not ÷6.

The same "check the prime factors separately" idea gives rules for 12 (÷3 and ÷4), 15 (÷3 and ÷5), etc.