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# Divisibility

## Divisibility Rules

Divisor | Divisibility condition | Examples |
---|---|---|

1 | No special condition. Any integer is divisible by 1. | 2 is divisible by 1. |

2 | The last digit is even (0, 2, 4, 6, or 8). | 1,294: 4 is even. |

3 | Sum the digits. If the result is divisible by 3, then the original number is divisible by 3. | 405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3. |

4 | Examine the last two digits. | 40832: 32 is divisible by 4. |

If the tens digit is even, the ones digit must be 0, 4, or 8. | ||

If the tens digit is odd, the ones digit must be 2 or 6. | 40832: 3 is odd, and the last digit is 2. | |

Twice the tens digit, plus the ones digit. | 40832: 2 × 3 + 2 = 8, which is divisible by 4. | |

5 | The last digit is 0 or 5. | 495: the last digit is 5. |

6 | It is divisible by 2 and by 3. | 1,458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6. |

7 | Form the alternating sum of blocks of three from right to left. | 1,369,851: 851 − 369 + 1 = 483 = 7 × 69 |

Subtract 2 times the last digit from the rest. (Works because 21 is divisible by 7.) | 483: 48 − (3 × 2) = 42 = 7 × 6. | |

Or, add 5 times the last digit to the rest. (Works because 49 is divisible by 7.) | 483: 48 + (3 × 5) = 63 = 7 × 9. | |

Or, add 3 times the first digit to the next. (This works because 10a + b − 7a = 3a + b − last number has the same remainder) | 483: 4×3 + 8 = 20 remainder 6, 6×3 + 3 = 21. | |

Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, -1, -3, -2 (repeating for digits beyond the hundred-thousands place). Then sum the results. | 483595: (4 × (-2)) + (8 × (-3)) + (3 × (-1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7. | |

8 | If the hundreds digit is even, examine the number formed by the last two digits. | 26.01666667 |

If the hundreds digit is odd, examine the number obtained by the last two digits plus 4. | 352: 52 + 4 = 56. | |

Add the last digit to twice the rest. | 56: (5 × 2) + 6 = 16. | |

Examine the last three digits. | 34152: Examine divisibility of just 152: 19 × 8 | |

Add four times the hundreds digit to twice the tens digit to the ones digit. | 34152: 4 × 1 + 5 × 2 + 2 = 16 | |

9 | Sum the digits. If the result is divisible by 9, then the original number is divisible by 9. | 2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9. |

10 | The last digit is 0. | 130: the last digit is 0. |

11 | Form the alternating sum of the digits. | 918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22. |

Add the digits in blocks of two from right to left.[1] | 627: 6 + 27 = 33. | |

Subtract the last digit from the rest. | 627: 62 − 7 = 55. | |

Add the last digit to the hundredth place (add 10 times the last digit to the rest). | 627: 62 + 70 = 132. | |

If the number of digits is even, add the first and subtract the last digit from the rest. | 918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11 | |

If the number of digits is odd, subtract the first and last digit from the rest. | 14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 11 | |

12 | It is divisible by 3 and by 4. | 324: it is divisible by 3 and by 4. |

Subtract the last digit from twice the rest. | 324: 32 × 2 − 4 = 60. | |

13 | Form the alternating sum of blocks of three from right to left. | 2,911,272: −2 + 911 − 272 = 637 |

Add 4 times the last digit to the rest. | 637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13. | |

Subtract 9 times the last digit from the rest. | 637: 63 - 63 = 0. | |

14 | It is divisible by 2 and by 7. | 224: it is divisible by 2 and by 7. |

Add the last two digits to twice the rest. The answer must be divisible by 14. | 364: 3 × 2 + 64 = 70. | |

1764: 17 × 2 + 64 = 98. | ||

15 | It is divisible by 3 and by 5. | 390: it is divisible by 3 and by 5. |

16 | If the thousands digit is even, examine the number formed by the last three digits. | 254,176: 176. |

If the thousands digit is odd, examine the number formed by the last three digits plus 8. | 3,408: 408 + 8 = 416. | |

Add the last two digits to four times the rest. | 176: 1 × 4 + 76 = 80. | |

1168: 11 × 4 + 68 = 112. | ||

Examine the last four digits. | 157,648: 7,648 = 478 × 16. | |

17 | Subtract 5 times the last digit from the rest. | 221: 22 − 1 × 5 = 17. |

18 | It is divisible by 2 and by 9. | 342: it is divisible by 2 and by 9. |

19 | Add twice the last digit to the rest. | 437: 43 + 7 × 2 = 57. |

20 | It is divisible by 10, and the tens digit is even. | 360: is divisible by 10, and 6 is even. |

If the number formed by the last two digits is divisible by 20. | 480: 80 is divisible by 20. |

|| Number Systems || Divisibility Rules || Remainder Theorem || Series and Progressions || Fractions and Decimal fractions || Solved Examples || Practice Questions ||

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Divisibility
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