Divisibility Rules and Process
Divisibility rules help quickly determine if a number is divisible by another without performing full division.
1. Divisibility Rule of 2
Rule:
A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
Example:
- 128 → Last digit is 8 (divisible ✅)
- 357 → Last digit is 7 (not divisible ❌)
2. Divisibility Rule of 3
Rule:
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example:
- 162 → (divisible ✅)
- 247 → (not divisible ❌)
3. Divisibility Rule of 4
Rule:
A number is divisible by 4 if its last two digits are divisible by 4.
Example:
- 316 → Last two digits 16 (16 ÷ 4 = 4 ✅)
- 527 → Last two digits 27 (not divisible ❌)
4. Divisibility Rule of 5
Rule:
A number is divisible by 5 if its last digit is 0 or 5.
Example:
- 225 → Last digit is 5 (✅)
- 348 → Last digit is 8 (❌)
5. Divisibility Rule of 6
Rule:
A number is divisible by 6 if it is divisible by both 2 and 3.
Example:
- 132 → Even (divisible by 2) and 1+3+2 = 6 (divisible by 3) ✅
- 215 → Not even (not divisible by 2) ❌
6. Divisibility Rule of 7
Rule:
Double the last digit, subtract it from the remaining number. If the result is divisible by 7, the number is also divisible.
Example:
- 343 → (divisible ✅)
- 259 → (divisible ✅)
7. Divisibility Rule of 8
Rule:
A number is divisible by 8 if its last three digits are divisible by 8.
Example:
- 512 → Last three digits 512 (512 ÷ 8 = 64 ✅)
- 974 → Last three digits 974 (not divisible ❌)
8. Divisibility Rule of 9
Rule:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
- 729 → (divisible ✅)
- 345 → (not divisible ❌)
9. Divisibility Rule of 10
Rule:
A number is divisible by 10 if its last digit is 0.
Example:
- 890 → (✅)
- 267 → (❌)
10. Divisibility Rule of 11
Rule:
A number is divisible by 11 if the difference between the sum of its alternate digits is 0 or a multiple of 11.
Example:
- 121 →
- 462 →
11. Divisibility Rule of 12
Rule:
A number is divisible by 12 if it is divisible by both 3 and 4.
Example:
- 312 → (divisible by 3 ✅) and 12 (last two digits) divisible by 4 ✅
- 154 → Not divisible by 3 ❌
Summary Table
| Number | Divisibility Rule |
|---|---|
| 2 | Last digit 0, 2, 4, 6, 8. |
| 3 | Sum of digits divisible by 3. |
| 4 | Last two digits divisible by 4. |
| 5 | Last digit 0 or 5. |
| 6 | Divisible by both 2 and 3. |
| 7 | Double the last digit and subtract from the rest, result divisible by 7. |
| 8 | Last three digits divisible by 8. |
| 9 | Sum of digits divisible by 9. |
| 10 | Last digit 0. |
| 11 | Difference between sum of alternate digits is 0 or multiple of 11. |
| 12 | Divisible by both 3 and 4. |
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