Stupid
Divisibility Tricks
Marc Renault
The article Stupid Divisibility Tricks: 101 Ways to Stupefy Your Friends appeared in Math Horizons, November 2006.
PowerPoint slides for a presentation at MathFest in August, 2006.
Stupid Divisibility
Tricks Summary
Marc Renault
Suppose you have a number n that you wish to check for divisibility by d.
Ending digits: the number of ending digits of n to check for divisibility by d.
Add blocks: take the digits of n in blocks of the given length, and add them (or add and subtract alternately) starting from the right. d | n iff d divides the result.
Right trim 1: remove the rightmost digit of n, multiply it by the number in the column, and add that to the remaining digits of n. d | n iff d divides the result.
Right trim 2: like above, but remove the two rightmost digits of n.
Left trim 1: remove the leftmost digit of n, multiply it by the given number, shift in two places to the right, and add to the remaining digits of n. d | n iff d divides the result.
Use other tricks: apply divisibility tests for smaller divisors.
|
d |
end digits |
add blocks |
right trim 1 |
right trim 2 |
left trim 1 |
use other tricks |
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2 |
1 |
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3 |
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1, 2, 3 |
1, -2 |
1, -2 |
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4 |
2 |
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5 |
1 |
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6 |
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2 3 |
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7 |
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±3 |
5, -2 |
4, -3 |
2 |
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8 |
3 |
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9 |
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1, 2, 3 |
1 |
1 |
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10 |
1 |
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2 5 |
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11 |
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±1, 2, ±3 |
-1 |
1 |
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12 |
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3 4 |
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13 |
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±3 |
4, -9 |
3, -10 |
-4 |
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14 |
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2 |
2 7 |
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15 |
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3 5 |
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16 |
4 |
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17 |
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-5 |
8, -9 |
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18 |
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2 9 |
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19 |
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2 |
4 |
5 |
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20 |
2 |
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4 5 |
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21 |
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-2 |
4 |
-5 |
3 7 |
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22 |
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2 11 |
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23 |
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7 |
3, -20 |
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24 |
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8 3 |
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25 |
2 |
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26 |
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2 13 |
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27 |
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3 |
-8 |
10 |
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28 |
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4 7 |
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29 |
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3 |
-20 |
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30 |
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2 3 5 |
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31 |
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-3 |
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32 |
5 |
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4 |
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33 |
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2 |
10 |
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1 |
3 11 |
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34 |
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-2 |
2 17 |
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35 |
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-5 |
5 7 |
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36 |
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4 9 |
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37 |
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3 |
-11 |
10 |
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38 |
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2 19 |
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39 |
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4 |
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3 13 |
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40 |
3 |
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8 5 |
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41 |
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-4 |
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42 |
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2 3 7 |
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43 |
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-30 |
40, -3 |
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44 |
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4 11 |
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45 |
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9 5 |
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46 |
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2 23 |
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47 |
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8 |
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48 |
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4 |
16 3 |
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49 |
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5 |
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2 |
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50 |
2 |
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2 25 |
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51 |
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-5 |
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-2 |
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52 |
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-4 |
4 13 |
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53 |
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-9 |
-6 |
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54 |
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2 27 |
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55 |
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5 11 |
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56 |
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8 7 |
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57 |
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40 |
4 |
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3 19 |
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58 |
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2 29 |
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59 |
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6 |
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60 |
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4 3 5 |
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61 |
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-6 |
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62 |
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2 31 |
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63 |
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9 7 |
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64 |
6 |
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65 |
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5 13 |
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66 |
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2 3 11 |
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67 |
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-20 |
-2 |
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68 |
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4 17 |
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69 |
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7 |
-20 |
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3 23 |
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70 |
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2 5 7 |
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71 |
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-7 |
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72 |
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8 9 |
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73 |
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±4 |
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74 |
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2 37 |
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75 |
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3 25 |
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76 |
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4 19 |
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77 |
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±3 |
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-10 |
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7 11 |
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78 |
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2 3 13 |
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79 |
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8 |
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80 |
4 |
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16 5 |
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81 |
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-8 |
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82 |
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2 41 |
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83 |
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25 |
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84 |
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4 3 7 |
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85 |
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5 17 |
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86 |
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2 43 |
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87 |
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-20 |
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3 29 |
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88 |
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8 11 |
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89 |
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9, -80 |
-8 |
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90 |
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2 9 5 |
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91 |
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±3 |
-9 |
-10 |
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7 13 |
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92 |
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4 23 |
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93 |
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40 |
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3 31 |
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94 |
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2 47 |
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95 |
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5 |
5 19 |
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96 |
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4 |
32 3 |
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97 |
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3 |
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98 |
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2 |
2 49 |
|
99 |
|
2 |
10 |
1 |
1 |
9 11 |
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100 |
2 |
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4 25 |