Divisibility rules help us work out whether a number is exactly divisible by other numbers (i.e. there is no remainder).

The rules are shortcuts for finding out whether numbers are exactly divisible without doing division calculations. Some of these rules along with examples are illustrated below:

Rule: If the sum of the digits is a multiple of 3 | ||

Number | Divisible? | Why? |

405 | Yes | 4 + 0 + 5 = 9 (9 is a multiple of 3) |

381 | Yes | 3 + 8 + 1 = 12 (12 is a multiple of 3) |

928 | No | 9 + 2 + 8 = 19 (19 is not a multiple of 3) |

4,616 | No | 4 + 6 + 1 + 6 = 17 (17 is not a multiple of 3) |

Helper: The multiples of 3 include... |

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 |

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Rule: If the last two digits are a multiple of 4 (or if the last two digits are 00) |
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Number | Divisible? | Why? |

348 | Yes | 48 is a multiple of 4 |

27,616 | Yes | 16 is a multiple of 4 |

8,514 | No | 14 is not a multiple of 4 |

722 | No | 22 is not a multiple of 4 |

1,200 | Yes | The last two digits are 00 200 is a multiple of 4 |

Helper: The multiples of 4 include... |

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 |

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Rule: If it is divisible by 2 and by 3 | ||

Number | Divisible? | Why? |

5,106 | Yes | The last digit is a 2 (it is a multiple of 2 ) and... 5 + 1 + 0 + 6 = 12 (12 is a multiple of 3) |

636 | Yes | The last digit is a 6 (it is a multiple of 2 ) and... 6 + 3 + 6 = 15 (15 is a multiple of 3) |

5,912 | No | The last digit is a 2 (it is a multiple of 2 ) 5 + 9 + 1 + 2 = 17 (17 is |

508 | No | The last digit is a 8 (it is a multiple of 2 ) 5 + 0 + 8 = 13 (13 is |

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Rule: If the sum of the digits are a multiple of 9 | ||

Number | Divisible? | Why? |

7,686 | Yes | 7 + 6 + 8 + 6 = 27 (27 is a multiple of 9) |

252 | Yes | 2 + 5 + 2 = 9 (9 is a multiple of 9) |

883 | No | 8 + 8 + 3 = 19 (19 is not a multiple of 9) |

5,105 | No | 5 + 1 + 0 + 5 = 11 (11 is not a multiple of 9) |

Helper: The multiples of 9 include... |

9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 |

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