Number System — Divisibility Rules (Quick Notes & Shortcuts)
Concise rules, formulas in brackets, step-by-step checks, and shortcuts — perfect for aptitude practice.
Number system Can be divided into different types
A number is divisible by another if division leaves no remainder. Divisibility rules let you test divisibility quickly without performing full division.
Rules (Step-by-step with examples)
-
Divisible by 2
Rule: Last digit is even. [Last digit ∈ {0,2,4,6,8}]
Example: 1,234 → last digit 4 → divisible by 2. -
Divisible by 3
Rule: Sum of digits divisible by 3. [Sum of digits]
Example: 1,425 → 1+4+2+5 = 12 → 12 divisible by 3 → number divisible by 3. -
Divisible by 4
Rule: Last 2 digits divisible by 4. [Number formed by last two digits]
Example: 7,316 → last 2 digits 16 → 16 ÷ 4 = 4 → divisible by 4. -
Divisible by 5
Rule: Last digit is 0 or 5. [Last digit]
Example: 2,345 → last digit 5 → divisible by 5. -
Divisible by 6
Rule: Divisible by both 2 and 3. (i.e., [Even] and [Sum of digits divisible by 3])
Example: 1,236 → even and 1+2+3+6=12 (÷3) → divisible by 6. -
Divisible by 7
Rule (common quick test): Double last digit and subtract from remaining truncated number. Repeat until small. If result divisible by 7 → original divisible.
Example: 203 → 20 − 2×3 = 20 − 6 = 14 → 14 ÷ 7 = 2 → divisible by 7. -
Divisible by 8
Rule: Last 3 digits divisible by 8. [Number formed by last three digits]
Example: 54,280 → last 3 digits 280 → 280 ÷ 8 = 35 → divisible by 8. -
Divisible by 9
Rule: Sum of digits divisible by 9. [Sum of digits]
Example: 2,343 → 2+3+4+3 = 12 → not divisible by 9. -
Divisible by 10
Rule: Last digit is 0. [Last digit]
Example: 5,930 → last digit 0 → divisible by 10. -
Divisible by 11
Rule: Find the difference between the sum of digits at odd positions and the sum at even positions (counting from right or left consistently). If result is 0 or divisible by 11 → divisible by 11. [Σ(odd pos) − Σ(even pos)]
Example: 2,530 → (2 + 3) − (5 + 0) = 5 − 5 = 0 → divisible by 11. -
Divisible by 12
Rule: Divisible by both 3 and 4. [Check: Sum of digits divisible by 3 & last two digits divisible by 4]
Shortcuts & Tips
- 3 & 9 rules are digit-sum based — great for quick elimination. [Sum of digits]
- Combine rules: To test divisibility by 24 → check divisible by 3 and 8 (or 4 and 6 depending on ease).
- 7, 13, 17, 19: There are subtraction/multiplication tricks — but they are less used in short aptitude questions.
- Even/odd checks remove many options quickly in multiple-choice tests.
- Memorize small multiples: Multiples of 7 up to 140, and powers of 2 up to 1024 help speed mental checks for 7 and 8.
Quick Reference Table
| Divisor | Rule | Shortcut (in brackets) |
|---|---|---|
| 2 | Last digit even | [Last digit] |
| 3 | Sum of digits ÷ 3 | [Sum of digits] |
| 4 | Last 2 digits ÷ 4 | [Last 2 digits] |
| 5 | Last digit 0 or 5 | [Last digit] |
| 6 | Divisible by 2 & 3 | [2 & 3] |
| 7 | Double last digit, subtract | [2×last digit] |
| 8 | Last 3 digits ÷ 8 | [Last 3 digits] |
| 9 | Sum of digits ÷ 9 | [Sum of digits] |
| 10 | Last digit 0 | [Last digit] |
| 11 | (Odd pos sum − Even pos sum) ÷ 11 | [Alternate sum diff] |
| 12 | Divisible by 3 & 4 | [3 & 4] |
3 Quick Practice Questions
- Is 5,832 divisible by 8? (Check last 3 digits)
- Is 7,329 divisible by 3 or 9? (Use sum of digits)
- Check divisibility by 11 for 27,391. (Alternate sum rule)
Answers (click to reveal)
1. 5,832 → last 3 digits 832. 832 ÷ 8 = 104 → Yes.
2. 7,329 → 7+3+2+9 = 21 → 21 ÷ 3 = 7 so divisible by 3. 21 ÷ 9 = 2 remainder 3 → Not divisible by 9.
3. 27,391 → (2+7+9) − (3+1) = 18 − 4 = 14 → 14 ÷ 11 = remainder → Not divisible by 11.
Divisibility Rules – Practice MCQs
Each question tests your knowledge of basic divisibility rules. Work step-by-step and apply the correct test.
Q1. Which of the following numbers is divisible by 2?
a) 1345 b) 2457 c) 8624 d) 9131
Answer: c) 8624
Explanation: Rule → A number is divisible by 2 if its last digit is even. Last digit of 8624 is 4 (even). Hence divisible.
Q2. Find the number divisible by 3.
a) 5112 b) 6421 c) 7327 d) 8455
Answer: a) 5112
Explanation: Rule → Divisible by 3 if sum of digits is divisible by 3.
Sum of 5112 = 5+1+1+2 = 9 → divisible by 3.
Q3. Which of the following is divisible by 4?
a) 4312 b) 7645 c) 2597 d) 9219
Answer: a) 4312
Explanation: Rule → Divisible by 4 if last two digits divisible by 4.
Last two digits = 12 → 12 ÷ 4 = 3. Divisible.
Q4. Which number is divisible by 5?
a) 9321 b) 1040 c) 6783 d) 2497
Answer: b) 1040
Explanation: Rule → Divisible by 5 if last digit is 0 or 5. Last digit of 1040 is 0. Hence divisible.
Q5. Which of the following is divisible by 6?
a) 2482 b) 3450 c) 4671 d) 9327
Answer: b) 3450
Explanation: Rule → Must be divisible by both 2 and 3.
Last digit of 3450 = 0 (even → divisible by 2).
Sum = 3+4+5+0 = 12 (divisible by 3). So divisible by 6.
Q6. Check divisibility by 7: Which number passes?
a) 322 b) 385 c) 224 d) 196
Answer: d) 196
Explanation: Rule → Double last digit and subtract.
196 → 19 − (2×6) = 19 − 12 = 7 → divisible by 7. So 196 divisible.
Q7. Which of these numbers is divisible by 8?
a) 4321 b) 9200 c) 7568 d) 1345
Answer: c) 7568
Explanation: Rule → Divisible by 8 if last 3 digits divisible by 8.
Last 3 digits = 568 → 568 ÷ 8 = 71. Divisible.
Q8. Which number is divisible by 9?
a) 5721 b) 6390 c) 8427 d) 9254
Answer: c) 8427
Explanation: Rule → Sum of digits divisible by 9.
8+4+2+7 = 21 → not divisible by 9. Wait, check again.
(Correction) 8427 → Sum = 21, not divisible. → Error? Let's correct:
Let’s re-pick: 6390 → 6+3+9+0 = 18 → divisible by 9.
Correct Answer: b) 6390.
Q9. Which of these is divisible by 10?
a) 7345 b) 9182 c) 4050 d) 6879
Answer: c) 4050
Explanation: Rule → Divisible by 10 if last digit is 0. Here, 4050 ends with 0.
Q10. Which number is divisible by 11?
a) 2530 b) 7139 c) 9182 d) 4567
Answer: b) 7139
Explanation: Rule → Difference between sum of odd and even place digits divisible by 11.
7+3 = 10, 1+9 = 10 → Difference = 0 → divisible.
Q11. Which number is divisible by 12?
a) 2460 b) 3152 c) 1287 d) 4540
Answer: a) 2460
Explanation: Rule → Must be divisible by both 3 and 4.
Sum = 2+4+6+0 = 12 (divisible by 3). Last 2 digits = 60 (divisible by 4). So divisible by 12.
Q12. Which number is divisible by 7?
a) 189 b) 201 c) 322 d) 441
Answer: d) 441
Explanation: 441 ÷ 7 = 63 (exact). Rule test → 44 − 2×1 = 42, 42 ÷ 7 = 6. Divisible.
Q13. Find the number divisible by 8.
a) 624 b) 735 c) 902 d) 1116
Answer: a) 624
Explanation: Last 3 digits = 624 → 624 ÷ 8 = 78. Hence divisible.
Q14. Which is divisible by 9?
a) 3339 b) 4510 c) 5728 d) 6147
Answer: d) 6147
Explanation: Sum = 6+1+4+7 = 18 → divisible by 9. Hence number divisible.
Q15. Which number is divisible by 11?
a) 462 b) 918 c) 715 d) 847
Answer: a) 462
Explanation: Odd pos sum = 4+2 = 6, Even pos sum = 6. Difference = 0 → divisible by 11.