User:Dhrm77/pages/Divisibility rule

Divisibility rules for numbers 1–100
From the current Divisibility rule page, it appears that there is a technique for determining the divisibility of a number that can be extended to all numbers. Let's consider any number 'n' written in base 10. Let's name the last digit 'y', and the rest of the number x, so that n = 10*x + y. To determine the divisibility of this number 'n' by a divisor 'd' we can recombine x and y to form a smaller number, while keeping it's divisibility intact. Then we can repeat the process until it becomes obvious if 'n' is divisible by 'd' or not. For example: To determine if 12345678 is divisible by 7, we use the expression 1x -2y:
 * 1234567-16 = 1234551
 * 123455 -2 = 123453
 * 12345 - 6 = 12339
 * 1233 - 18 - 1215
 * 121 - 10 = 111
 * 11 -2 = 9, and 9 is not a multiple of 7, so neither is 12345678

But if we pick n = 12345676:
 * 1234567-12 = 1234555
 * 123455 -10 = 123445
 * 12344 - 10 = 12334
 * 1233 - 8 - 1225
 * 122 - 10 = 112
 * 11 - 4 = 7, which is a multiple of 7, so 12345676 is a multiple of 7

Now, I wrote a small program to figure out similar rules for every divisor from 2 to 99. Technically there are an infinite number of ways to process the numbers, but only 1 or 2 that are most efficient, meaning that with each step the number is reduces as much as possible.

In most cases, I listed the best 2 methods.

Here are the results:

Divisibility by 2: 0x +1y is divisible by 2, i.e Last digit is divisible by 2

Divisibility by 3: 1x +1y is divisible by 3, i.e. The sum of all digits is divisible by 3

Divisibility by 4: 2x -1y is divisible by 4 2x +1y is divisible by 4

Divisibility by 5: 0x +1y is divisible by 5, i.e Last digit is divisible by 5

Divisibility by 6: 2x -1y is divisible by 6 4x +1y is divisible by 6

Divisibility by 7: 1x -2y is divisible by 7 3x +1y is divisible by 7

Divisibility by 8: 2x +1y is divisible by 8 2x -3y is divisible by 8

Divisibility by 9: 1x +1y is divisible by 9, i.e. The sum of all digits is divisible by 9

Divisibility by 10: 0x +1y is divisible by 10, i.e. Last digit is divisible by 10, i.e, last digit is zero.

Divisibility by 11: 1x -1y is divisible by 11 6x +5y is divisible by 11

Divisibility by 12: 2x -1y is divisible by 12 2x +5y is divisible by 12

Divisibility by 13: 3x -1y is divisible by 13 1x +4y is divisible by 13

Divisibility by 14: 4x -1y is divisible by 14 2x +3y is divisible by 14

Divisibility by 15: 5x -1y is divisible by 15 5x +2y is divisible by 15

Divisibility by 16: 2x -3y is divisible by 16 6x -1y is divisible by 16

Divisibility by 17: 3x +2y is divisible by 17 1x -5y is divisible by 17

Divisibility by 18: 8x -1y is divisible by 18 4x -5y is divisible by 18

Divisibility by 19: 1x +2y is divisible by 19 9x -1y is divisible by 19

Divisibility by 21: 1x -2y is divisible by 21 8x +5y is divisible by 21

Divisibility by 22: 2x +9y is divisible by 22 4x +7y is divisible by 22

Divisibility by 23: 3x -2y is divisible by 23 1x +7y is divisible by 23

Divisibility by 24: 2x +5y is divisible by 24 2x -7y is divisible by 24

Divisibility by 25: 5x -2y is divisible by 25 5x +3y is divisible by 25

Divisibility by 26: 2x -5y is divisible by 26 4x +3y is divisible by 26

Divisibility by 27: 7x -2y is divisible by 27 4x -5y is divisible by 27

Divisibility by 28: 2x +3y is divisible by 28 6x -5y is divisible by 28

Divisibility by 29: 1x +3y is divisible by 29 9x -2y is divisible by 29

Divisibility by 31: 1x -3y is divisible by 31 9x +4y is divisible by 31

Divisibility by 32: 2x -3y is divisible by 32 6x +7y is divisible by 32

Divisibility by 33: 4x +7y is divisible by 33 1x +10y is divisible by 33

Divisibility by 34: 4x -3y is divisible by 34 2x +7y is divisible by 34

Divisibility by 35: 5x -3y is divisible by 35 5x +4y is divisible by 35

Divisibility by 36: 2x -7y is divisible by 36 2x +11y is divisible by 36

Divisibility by 37: 3x +4y is divisible by 37 7x -3y is divisible by 37

Divisibility by 38: 8x -3y is divisible by 38 6x -7y is divisible by 38

Divisibility by 39: 1x +4y is divisible by 39 8x -7y is divisible by 39

Divisibility by 41: 1x -4y is divisible by 41 9x +5y is divisible by 41

Divisibility by 42: 8x +5y is divisible by 42 4x +13y is divisible by 42

Divisibility by 43: 3x -4y is divisible by 43 7x +5y is divisible by 43

Divisibility by 44: 2x +9y is divisible by 44 6x +5y is divisible by 44

Divisibility by 45: 5x -4y is divisible by 45 5x -13y is divisible by 45

Divisibility by 46: 4x +5y is divisible by 46 2x -9y is divisible by 46

Divisibility by 47: 3x +5y is divisible by 47 7x -4y is divisible by 47

Divisibility by 48: 2x +5y is divisible by 48 2x -19y is divisible by 48

Divisibility by 49: 1x +5y is divisible by 49 9x -4y is divisible by 49

Divisibility by 51: 1x -5y is divisible by 51 8x +11y is divisible by 51

Divisibility by 52: 2x -5y is divisible by 52 6x +11y is divisible by 52

Divisibility by 53: 3x -5y is divisible by 53 7x +6y is divisible by 53

Divisibility by 54: 4x -5y is divisible by 54 2x +11y is divisible by 54

Divisibility by 55: 5x +6y is divisible by 55 5x -16y is divisible by 55

Divisibility by 56: 6x -5y is divisible by 56 2x -11y is divisible by 56

Divisibility by 57: 7x -5y is divisible by 57 4x -11y is divisible by 57

Divisibility by 58: 8x -5y is divisible by 58 6x -11y is divisible by 58

Divisibility by 59: 1x +6y is divisible by 59 9x -5y is divisible by 59

Divisibility by 61: 1x -6y is divisible by 61 9x +7y is divisible by 61

Divisibility by 62: 8x +7y is divisible by 62 6x +13y is divisible by 62

Divisibility by 63: 4x +13y is divisible by 63 1x +19y is divisible by 63

Divisibility by 64: 6x +7y is divisible by 64 2x +13y is divisible by 64

Divisibility by 65: 5x -6y is divisible by 65 5x +7y is divisible by 65

Divisibility by 66: 4x +7y is divisible by 66 2x -13y is divisible by 66

Divisibility by 67: 3x +7y is divisible by 67 7x -6y is divisible by 67

Divisibility by 68: 2x +7y is divisible by 68 6x -13y is divisible by 68

Divisibility by 69: 1x +7y is divisible by 69 8x -13y is divisible by 69

Divisibility by 71: 1x -7y is divisible by 71 9x +8y is divisible by 71

Divisibility by 72: 2x -7y is divisible by 72 2x +29y is divisible by 72

Divisibility by 73: 3x -7y is divisible by 73 7x +8y is divisible by 73

Divisibility by 74: 4x -7y is divisible by 74 2x +15y is divisible by 74

Divisibility by 75: 5x -7y is divisible by 75 5x +8y is divisible by 75

Divisibility by 76: 6x -7y is divisible by 76 2x -15y is divisible by 76

Divisibility by 77: 3x +8y is divisible by 77 4x -15y is divisible by 77

Divisibility by 78: 8x -7y is divisible by 78 4x -23y is divisible by 78

Divisibility by 79: 1x +8y is divisible by 79 9x -7y is divisible by 79

Divisibility by 81: 1x -8y is divisible by 81 8x +17y is divisible by 81

Divisibility by 82: 8x +9y is divisible by 82 6x +17y is divisible by 82

Divisibility by 83: 3x -8y is divisible by 83 7x +9y is divisible by 83

Divisibility by 84: 2x +17y is divisible by 84 2x -25y is divisible by 84

Divisibility by 85: 5x -8y is divisible by 85 5x +9y is divisible by 85

Divisibility by 86: 4x +9y is divisible by 86 2x -17y is divisible by 86

Divisibility by 87: 7x -8y is divisible by 87 4x -17y is divisible by 87

Divisibility by 88: 2x +9y is divisible by 88 6x -17y is divisible by 88

Divisibility by 89: 1x +9y is divisible by 89 9x -8y is divisible by 89

Divisibility by 91: 1x -9y is divisible by 91 9x +10y is divisible by 91

Divisibility by 92: 2x -9y is divisible by 92 6x +19y is divisible by 92

Divisibility by 93: 7x +10y is divisible by 93 4x +19y is divisible by 93

Divisibility by 94: 4x -9y is divisible by 94 2x +19y is divisible by 94

Divisibility by 95: 5x -9y is divisible by 95 5x -28y is divisible by 95

Divisibility by 96: 2x -19y is divisible by 96 2x +29y is divisible by 96

Divisibility by 97: 3x +10y is divisible by 97 7x -9y is divisible by 97

Divisibility by 98: 8x -9y is divisible by 98 6x -19y is divisible by 98

Divisibility by 99: 1x +10y is divisible by 99 8x -19y is divisible by 99

I realize that this can be considered Original Work, but at the same time, this is really simple and easily verifiable.