Quick Concept Summary
Divisibility rules let you check if a number is divisible by another without performing actual division. In exams, they speed up simplification, LCM/HCF, and prime factorisation problems.
Divisibility Rules — All in One Table
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 | 348 ✓ (ends in 8) |
| 3 | Sum of all digits divisible by 3 | 123 → 1+2+3=6 ✓ |
| 4 | Last two digits form a number divisible by 4 | 1312 → 12÷4=3 ✓ |
| 5 | Last digit is 0 or 5 | 395 ✓ (ends in 5) |
| 6 | Divisible by both 2 AND 3 | 126 → even, 1+2+6=9 ✓ |
| 7 | Double last digit, subtract from rest; result divisible by 7 | 343 → 34−6=28 ✓ |
| 8 | Last three digits divisible by 8 | 4120 → 120÷8=15 ✓ |
| 9 | Sum of all digits divisible by 9 | 729 → 7+2+9=18 ✓ |
| 10 | Last digit is 0 | 1230 ✓ |
| 11 | (Odd-position sum) − (Even-position sum) = 0 or multiple of 11 | 1331 → (1+3)−(3+1)=0 ✓ |
| 12 | Divisible by both 3 AND 4 | 144 → 1+4+4=9 ✓, 44÷4=11 ✓ |
| 15 | Divisible by both 3 AND 5 | 360 → 3+6+0=9 ✓, ends in 0 ✓ |
| 18 | Divisible by both 2 AND 9 | 162 → even ✓, 1+6+2=9 ✓ |
| 25 | Last two digits form 00, 25, 50, or 75 | 1275 → ends in 75 ✓ |
Worked Examples
Example 1 — Divisibility by 3 and 9
Q: Is 45,612 divisible by 3? By 9?
Digit sum = 4+5+6+1+2 = 18
- 18 ÷ 3 = 6 → Yes, divisible by 3 ✓
- 18 ÷ 9 = 2 → Yes, divisible by 9 ✓
Example 2 — Divisibility by 11
Q: Is 83,512 divisible by 11?
Position the digits: 8(1), 3(2), 5(3), 1(4), 2(5)
- Odd positions (1st, 3rd, 5th): 8 + 5 + 2 = 15
- Even positions (2nd, 4th): 3 + 1 = 4
- Difference = 15 − 4 = 11
11 is divisible by 11 → Yes, 83,512 is divisible by 11 ✓
Example 3 — Divisibility by 7
Q: Is 1,001 divisible by 7?
Apply rule repeatedly:
1001 → 100 − (1×2) = 100 − 2 = 98
98 → 9 − (8×2) = 9 − 16 = −7
−7 is divisible by 7 → Yes, 1001 is divisible by 7 ✓
(In fact 1001 = 7 × 143)
Example 4 — Finding Missing Digit
Q: In 4_8, the missing digit makes the number divisible by 9. What is the missing digit?
Digit sum = 4 + ? + 8 = 12 + ?
For divisibility by 9: 12 + ? must be a multiple of 9.
Next multiple of 9 after 12 is 18 → ? = 6
Check: 468 → 4+6+8 = 18, 18÷9 = 2 ✓
Example 5 — Divisibility by 4 and 8
Q: Is 87,416 divisible by 4? By 8?
- Last two digits: 16. 16 ÷ 4 = 4 → Divisible by 4 ✓
- Last three digits: 416. 416 ÷ 8 = 52 → Divisible by 8 ✓
Example 6 — Composite Divisibility
Q: Which of the following is divisible by 6: 324, 512, 630, 714?
Must be even AND digit sum divisible by 3:
| Number | Even? | Digit sum | Sum ÷ 3? |
|---|---|---|---|
| 324 | ✓ | 9 | ✓ |
| 512 | ✓ | 8 | ✗ |
| 630 | ✓ | 9 | ✓ |
| 714 | ✓ | 12 | ✓ |
324, 630, and 714 are divisible by 6.
Shortcuts and Tricks
Trick 1 — Divisibility by 6, 12, 15, 18 = combined rules
| To check | Check both |
|---|---|
| Div by 6 | Div by 2 AND div by 3 |
| Div by 12 | Div by 3 AND div by 4 |
| Div by 15 | Div by 3 AND div by 5 |
| Div by 18 | Div by 2 AND div by 9 |
Trick 2 — Casting out nines (remainder by 9)
The remainder when a number is divided by 9 equals the remainder when its digit sum is divided by 9.
Example: 6,735 ÷ 9 → digit sum = 6+7+3+5 = 21, 21 mod 9 = 3. So 6735 mod 9 = 3.
Trick 3 — Divisibility by 11 quick pattern
For a 3-digit number ABC: it's divisible by 11 if A − B + C = 0 or 11.
Example: 121 → 1 − 2 + 1 = 0 ✓. Example: 352 → 3 − 5 + 2 = 0 ✓.
Trick 4 — Missing digit in a divisibility question
Always: digit sum needed = nearest multiple of 3 or 9 above the current partial sum. Subtract to find the missing digit.
Exam Pattern
TNPSC: 1–2 questions per paper. Usually ask: "Which number is divisible by X?" or "What digit completes divisibility by Y?"
SSC/Bank PO: Up to 3 questions. Include finding missing digits, combined divisibility (by 6, 12, 18), and sometimes divisibility by 7 or 11.
Placement: Often in quantitative reasoning sections. Quick elimination using divisibility rules saves time in large MCQ sets.
Practice Questions
| # | Question | Answer |
|---|---|---|
| 1 | Is 7,482 divisible by 3? | Yes (7+4+8+2=21) |
| 2 | Is 5,236 divisible by 4? | Yes (last two: 36÷4=9) |
| 3 | Is 84,523 divisible by 9? | No (8+4+5+2+3=22) |
| 4 | What digit replaces ? in 6?2 to make it divisible by 3? | 1, 4, or 7 |
| 5 | Is 19,261 divisible by 11? | Yes ((1+2+1)−(9+6)=4−15=−11) |
| 6 | Which is divisible by 8: 1704, 1720, 1736? | All three (104, 120, 136 all ÷ 8) |
| 7 | Is 3,024 divisible by 6? | Yes (even, 3+0+2+4=9) |
| 8 | What smallest digit makes 52_4 divisible by 4? | 0 (last two: 04) |
| 9 | Is 1,234,567 divisible by 7? | No (apply rule: 123456 − 14 = 123442 → keep reducing) |
| 10 | Is 2,178 divisible by both 3 and 11? | Yes (2+1+7+8=18 ✓; (2+7)−(1+8)=0 ✓) |