and 11,12,13,17,19,23,29,31,37,41,43,47

Divisible by: | Test or Rule | Examples: |
---|---|---|

2 | The last digit is even (0,2,4,6,8) | 128 |

3 | The sum of the digits is divisible by 3 | Example :381 (3+8+1=12, and 12/3 = 4) |

4 | The last 2 digits are divisible by 4 | Example: 1312 is (12/4=3) |

5 | The last digit is 0 or 5 | 175 |

6 | The number is divisible by both 2 and 3 | Example: 114 (it is even, and 1+1+4=6 and 6/3 = 2) |

7 | If you double the last digit and subtract it from the rest of the number and the answer is: 0 or divisible by 7 | Example: 672 (Double 2 is 4, 67-4=63, and 63/7=9) |

8 | The last three digits are divisible by 8 | Example: 109816 (816/8=102) |

9 | The sum of the digits is divisible by 9 | Example: 1629 (1+6+2+9=18, and again, 1+8=9) |

10 | The number ends in 0 | 220 |

11 | If you sum every second digit and then subtract all other digits and the answer is: 0 or divisible by 11 | Example: 1364 ((3+4) - (1+6) =0) & 3729 ((7+9) - (3+2) = 11) |

12 | The number is divisible by both 3 and 4 | 648 (By 3? 6+4+8=18 and 18/3=6 By 4? 48/4=12) |

13 | Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. | Example: 50661-->5066+4=5070-->507+0=507-->50+28=78 and 78 is 6*13, so 50661 is divisible by 13. |

17 | Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. | Example: 3978-->397-5*8=357-->35-5*7=0. So 3978 is divisible by 17. |

19 | Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary | Example: 101156-->10115+2*6=10127-->1012+2*7=1026-->102+2*6=114 and 114=6*19, so 101156 is divisible by 19. |

23 | 3*23=69, ends in a 9, so ADD. Add 7 times the last digit to the remaining leading truncated number. If the result is divisible by 23, then so was the first number. Apply this rule over and over again as necessary. | Example: 17043-->1704+7*3=1725-->172+7*5=207 which is 9*23, so 17043 is also divisible by 23. |

29 | Add three times the last digit to the remaining leading truncated number. If the result is divisible by 29, then so was the first number. Apply this rule over and over again as necessary. | Example: 15689-->1568+3*9=1595-->159+3*5=174-->17+3*4=29, so 15689 is also divisible by 29. |

31 | Subtract three times the last digit from the remaining leading truncated number. If the result is divisible by 31, then so was the first number. Apply this rule over and over again as necessary. | Example: 7998-->799-3*8=775-->77-3*5=62 which is twice 31, so 7998 is also divisible by 31. |

37 | This is (slightly) more difficult, since it perforce uses a double-digit multiplier, namely eleven. People can usually do single digit multiples of 11, so we can use the same technique still. Subtract eleven times the last digit from the remaining leading truncated number. If the result is divisible by 37, then so was the first number. Apply this rule over and over again as necessary. | Example: 23384-->2338-11*4=2294-->229-11*4=185 which is five times 37, so 23384 is also divisible by 37. |

41 | Subtract four times the last digit from the remaining leading truncated number. If the result is divisible by 41, then so was the first number. Apply this rule over and over again as necessary. | Example: 30873-->3087-4*3=3075-->307-4*5=287-->28-4*7=0, remainder is zero and so 30873 is also divisible by 41. |

43 | Now it starts to get really difficult for most people, because the multiplier to be used is 13, and most people cannot recognise even single digit multiples of 13 at sight. You may want to make a little list of 13*N first. Nevertheless, for the sake of completeness, we will use the same method. Add thirteen times the last digit to the remaining leading truncated number. If the result is divisible by 43, then so was the first number. Apply this rule over and over again as necessary. | Example: 3182-->318+13*2=344-->34+13*4=86 which is recognisably twice 43, and so 3182 is also divisible by 43. |

47 | This too is difficult for most people, because the multiplier to be used is 14, and most people cannot recognise even single digit multiples of 14 at sight. You may want to make a little list of 14*N first. Nevertheless, for the sake of completeness, we will use the same method. Subtract fourteen times the last digit from the remaining leading truncated number. If the result is divisible by 47, then so was the first number. Apply this rule over and over again as necessary. | Example: 34827-->3482-14*7=3384-->338-14*4=282-->28-14*2=0 , remainder is zero and so 34827 is divisible by 47. |

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