Quick Divisibility Tests for 2, 3, 4, 5, 8, 9, 11

Instant Divisibility Checks

You can tell if a number is divisible by 2, 3, 4, 5, 8, 9, or 11 just by looking at its digits!

Divisible by 2

Rule: Last digit is even (0, 2, 4, 6, 8)

• 348 → last digit 8 (even) → YES ✓
• 567 → last digit 7 (odd) → NO ✗

Divisible by 3

Rule: Sum of digits is divisible by 3

• 246 → 2+4+6 = 12 → 12÷3 = 4 → YES ✓
• 457 → 4+5+7 = 16 → 16÷3 has remainder → NO ✗

Divisible by 4

Rule: Last TWO digits form a number divisible by 4

• 1236 → last two: 36 → 36÷4 = 9 → YES ✓
• 3418 → last two: 18 → 18÷4 has remainder → NO ✗
• Quick check: if tens digit is even, ones must be 0, 4, or 8
• If tens digit is odd, ones must be 2 or 6

Divisible by 5

Rule: Last digit is 0 or 5

• 385 → last digit 5 → YES ✓
• 2470 → last digit 0 → YES ✓
• 892 → last digit 2 → NO ✗

Divisible by 8

Rule: Last THREE digits form a number divisible by 8

• 12416 → last three: 416 → 416÷8 = 52 → YES ✓
• 5123 → last three: 123 → not divisible by 8 → NO ✗

Divisible by 9

Rule: Sum of digits is divisible by 9

• 4563 → 4+5+6+3 = 18 → 18÷9 = 2 → YES ✓
• 1234 → 1+2+3+4 = 10 → not divisible by 9 → NO ✗

Divisible by 11

Rule: Alternating sum of digits is 0 or divisible by 11

Start from right, alternately add and subtract:

• 1331 → 1 - 3 + 3 - 1 = 0 → YES ✓
• 2728 → 8 - 2 + 7 - 2 = 11 → YES ✓
• 1234 → 4 - 3 + 2 - 1 = 2 → NO ✗

Why These Work

Powers of 10 Pattern:

• 10 ≡ 0 (mod 2, 5)
• 100 ≡ 0 (mod 4)
• 1000 ≡ 0 (mod 8)
• 10 ≡ 1 (mod 3, 9) → digit sum works
• 10 ≡ -1 (mod 11) → alternating sum works

Quick Reference Table

Divisor	Test
2	Last digit even
3	Sum of digits ÷ 3
4	Last 2 digits ÷ 4
5	Last digit 0 or 5
6	Divisible by both 2 AND 3
8	Last 3 digits ÷ 8
9	Sum of digits ÷ 9
10	Last digit is 0
11	Alternating digit sum