This page is the result of my work to find out as many divisibility tricks as possible. I worked for days to come up with most of these, however I adopted just a few of them.
Throughout the table, a few different procedures will be used. They are listed below:
Procedure 1- Divide digits of the number into groups of a. Add all the groups together. If the sum is divisible by n, then the number is divisible by n.
Procedure 2- Divide digits of the number into groups of a. Add every other group together, then add the remaining groups together. Find the difference of the two sums and if that is divisible by n, then the number is divisible by n.
Examples of each appears below the table.
| If the number "Method" then number is divisible by n. | |
|---|---|
| n | Method |
| 2 | last digit is even |
| 3 | Procedure 1: a=1 |
| 4 | has its last 2 digits divisible by 4 |
| 5 | ends in 0 or 5 |
| 6 | is divisible by 2 and 3 |
| 7 | Procedure 2. a=3 |
| 8 | has its last 3 digits divisible by 8 |
| 9 | Procedure 1. a=1 |
| 10 | ends in 0 |
| 11 | Procedure 1. a=2 |
| 12 | is divisible by 3 and 4 |
| 13 | Procedure 2. a=3 |
| 14 | is divisible by 2 and 7 |
| 15 | is divisible by 3 and 5 |
| 16 | has its last 4 digits divisible by 16 |
| 17 | Procedure 2: a=8 |
| 18 | is divisible by 2 and 9 |
| 19 | Procedure 2: a=9 |
| 20 | last digit is zero and tens digit is even |
| 21 | is divisible by 3 and 7 |
| 22 | is divisible by 2 and 11 |
| 23 | Procedure 2: a=11 |
| 24 | is divisible by 3 and 8 |
| 25 | ends in 00, 25, 50, or 75 |
| 26 | is divisible by 2 and 13 |
| 27 | Procedure 1: a=3 |
| 28 | is divisible by 4 and 7 |
| 29 | Procedure 2: a=14 |
| 30 | is divisible by 3 and 10 |
| 31 | Procedure 1: a=15 |
| 32 | has its last 5 digits divisible by 32 |
| 33 | is divisible by 3 and 11 |
| 34 | is divisible by 2 and 17 |
| 35 | is divisible by 5 and 7 |
| 36 | is divisible by 4 and 9 |
| 37 | Procedure 1: a=3 |
| 38 | is divisible by 2 and 19 |
| 39 | is divisible by 3 and 13 |
| 40 | ends in zero, two digits to left of zero are divisible by 4 |
| 41 | Procedure 1: a=5 |
| 42 | is divisible by 2, 3, and 7 |
| 43 | Procedure 1: a=21 |
| 44 | is divisible by 4 and 11 |
| 45 | is divisible by 5 and 9 |
| 46 | is divisible by 2 and 23 |
| 47 | Procedure 2: a=23 |
| 48 | is divisible by 3 and 16 |
| 49 | Procedure 2: a=21 |
| 50 | ends in 00 or 50 |
| 51 | is divisible by 3 and 17 |
| 52 | is divisible by 4 and 13 |
| 53 | Procedure 1: a=13 |
| 54 | is divisible by 2 and 27 |
| 55 | is divisible by 5 and 11 |
| 56 | is divisible by 7 and 8 |
| 57 | is divisible by 3 and 19 |
| 58 | is divisible by 2 and 29 |
| 59 | Procedure 2: a=29 |
| 60 | is divisible by 3 and 20 |
| 61 | Procedure 2: a=30 |
| 62 | is divisible by 2 and 31 |
| 63 | is divisible by 7 and 9 |
| 64 | has its last 6 digits divisible by 64 |
| 65 | is divisible 5 and 13 |
| 66 | is divisible by 2, 3, and 11 |
| 67 | Procedure 1: a=33 |
| 68 | is divisible by 4 and 17 |
| 69 | is divisible by 3 and 23 |
| 70 | is divisible by 7 and 10 |
| 71 | Procedure 1: a=35 |
| 72 | is divisible by 8 and 9 |
| 73 | Procedure 1: a=4 |
| 74 | is divisible by 2 and 37 |
| 75 | is divisible by 3 and 25 |
| 76 | is divisible by 4 and 19 |
| 77 | is divisible by 7 and 11 |
| 78 | is divisible by 2, 3, and 13 |
| 79 | Procedure 1: a=13 |
| 80 | ends in zero, three digits to left of zero are divisible by 8 |
| 81 | Procedure 1: a=9 |
| 82 | is divisible by 2 and 41 |
| 83 | Procedure 1: a=41 |
| 84 | is divisible by 3, 4, and 7 |
| 85 | is divisible by 5 and 17 |
| 86 | is divisible by 2 and 43 |
| 87 | is divisible by 3 and 29 |
| 88 | is divisible by 8 and 11 |
| 89 | Procedure 1: a=44 |
| 90 | is divisible by 9 and 10 |
| 91 | is divisible by 7 and 13 |
| 92 | is divisible by 4 and 23 |
| 93 | is divisible by 3 and 31 |
| 94 | is divisible by 2 and 47 |
| 95 | is divisible by 5 and 19 |
| 96 | is divisible by 3 and 32 |
| 97 | Procedure 2: a=48 |
| 98 | is divisible by 2 and 49 |
| 99 | Procedure 1: a=2 |
| 100 | ends in 00 |
| 101 | Procedure 2: a=2 |
| 102 | is divisible by 2, 3 and 17 |
Examples of procedures 1 and 2:
One would use procedure 1 if he wanted to check for divisibility by 11. Say the number 257 513 575. First divide it into groups of 2 (since a=2) like this: 02, 57, 51, 35, 75. Then add them all up like this: 02 + 57 + 51 + 35 + 75 = 220. Since 220 is divisible by 11, 257 513 575 must be divisible by 11.
One would use procedure 2 if he wanted to check for divisibility by 7. Say the number 9 876 543 210. First divide it into groups of 3 (since a=3) like this: 009, 876, 543, 210. Find the sums of everyother group like this: 009 + 543 = 552 & 876 + 210 = 1086. Now find the difference id est: 1086 - 552 = 534. Since 534 is not divisible by 7, 9 876 543 210 is not divisible by 7.
Last Updated: 2009-05-02
The author, Marq Thompson, wished the content of this website to be uncopyrighted after his death.
