Divisibility Rules Practice & Tester
Test any number against all divisibility rules from 2 to 13 simultaneously and view step-by-step mathematical proofs.
Comprehensive Guide & Practice
Master the mathematical principles, formulas, and step-by-step methods with worked examples and FAQs.
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1. What are Divisibility Rules?
Divisibility rules are fast mental math shortcuts that allow you to determine whether a given integer is divisible by another integer without performing tedious pencil-and-paper long division.
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2. Complete Master Guide: Divisibility Rules from 2 to 13
| Divisor | The Divisibility Rule | Worked Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8). | 1,438 → Ends in 8 → Divisible. |
| 3 | The sum of the digits is divisible by 3. | 471 → 4+7+1 = 12 (12÷3=4) → Divisible. |
| 4 | The last two digits form a number divisible by 4. | 1,324 → 24 is divisible by 4 → Divisible. |
| 5 | Last digit is strictly 0 or 5. | 9,875 → Ends in 5 → Divisible. |
| 6 | Number is divisible by BOTH 2 and 3. | 792 → Even (divisible by 2) and 7+9+2=18 (divisible by 3) → Divisible. |
| 7 | Double the last digit and subtract from the rest. Result must be divisible by 7 (or 0). | 672 → 67 − (2×2) = 63 (63÷7=9) → Divisible. |
| 8 | The last three digits form a number divisible by 8. | 5,128 → 128 ÷ 8 = 16 → Divisible. |
| 9 | The sum of the digits is divisible by 9. | 2,853 → 2+8+5+3 = 18 (18÷9=2) → Divisible. |
| 10 | Last digit is 0. | 4,590 → Ends in 0 → Divisible. |
| 11 | Alternating sum of digits (add odd positions, subtract even positions) is 0 or a multiple of 11. | 2,728 → (2+2) − (7+8) = 4 − 15 = −11 (multiple of 11) → Divisible. |
| 12 | Number is divisible by BOTH 3 and 4. | 648 → Sum=18 (div by 3) and last two=48 (div by 4) → Divisible. |
| 13 | Multiply the last digit by 4 and add to the remaining truncated number. | 637 → 63 + (7×4) = 63 + 28 = 91 (91÷13=7) → Divisible. |
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3. Real-World Applications
- Simplifying Large Fractions: Quickly spot common factors without trial division.
- Barcodes & Check Digits: ISBN book codes and UPC barcodes use modulo 10 and modulo 11 checksum algorithms.
- Computer Architecture & Bit Shifting: Divisibility by powers of 2 (2, 4, 8, 16) corresponds directly to bitwise right-shift operations.