Divisibility Rules

A number is divisible by d if the remainder is 0 when divided by d (equivalently, d is a factor). Divisibility rules exploit the structure of our base-10 number system to check divisibility from the digits alone — much faster than actual division.

The rules work because every number can be written as a sum: 347 = 3×100 + 4×10 + 7. If you know how 100 and 10 behave modulo d, you can reduce the problem to just the digits.

• Rule for 2: Last digit is 0, 2, 4, 6, or 8
• Rule for 5: Last digit is 0 or 5
• Rule for 10: Last digit is 0

• Divisible by 3: Sum of all digits is divisible by 3. Example: 528 → 5+2+8=15, divisible by 3 ✓
• Divisible by 9: Sum of all digits is divisible by 9. Example: 729 → 7+2+9=18 ✓
• Divisible by 4: Last two digits form a number divisible by 4. Example: 1,316 → 16÷4=4 ✓
• Divisible by 8: Last three digits form a number divisible by 8. Example: 3,512 → 512÷8=64 ✓
• Divisible by 6: Divisible by BOTH 2 AND 3 (even number with digit sum divisible by 3)

Rule for 7 (less commonly memorised): Subtract twice the last digit from the remaining number. Repeat until small. If result is 0 or 7, divisible by 7. Example: 203 → 20 − 2×3 = 14 ✓

Rule for 11: Alternately add and subtract digits. If result is 0 or divisible by 11, the number is too. Example: 918,082 → 9−1+8−0+8−2 = 22 (divisible by 11) ✓

Rule for 12: Divisible by both 3 AND 4 simultaneously. Check the digit sum rule (÷3) AND the last-two-digits rule (÷4).

• Applying the 3-rule to check divisibility by 6: You need BOTH the 2-rule AND the 3-rule for 6. A number like 33 has digit sum 6 (÷3) but is odd (not ÷2), so it's not divisible by 6.
• Using the last digit for divisibility by 4: The rule for 4 requires the LAST TWO digits, not just the last one. 36 is not the same as 136 (136÷4=34 ✓, but testing last digit 6÷4 is wrong).
• Forgetting digit sum iteration: For large numbers, keep summing digits until you get a single number you can easily test.

Group the rules by difficulty:

• Easy (last digit only): 2 (even), 5 (0 or 5), 10 (0)
• Digit sum rules: 3 and 9 (sum divisible by 3 or 9)
• Last 2 digits: 4 and 25 (last two digits divisible)
• Compound rules: 6 = 2 AND 3; 12 = 3 AND 4; 15 = 3 AND 5; 18 = 2 AND 9

Use SolveCalc's Divisibility Checker to test any number against multiple divisors simultaneously.

• Rule for 8: Last three digits divisible by 8. Example: 1,720 → 720 ÷ 8 = 90 ✓
• Rule for 9: Sum of digits divisible by 9. Example: 729 → 7+2+9 = 18 ÷ 9 = 2 ✓
• Rule for 10: Last digit is 0. Example: 950 ✓
• Rule for 11: Alternating sum of digits is divisible by 11. Example: 1,331 → (1+3) − (3+1) = 4−4 = 0 ✓ (0 is divisible by 11)