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Math

Divisibility Tricks

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.
nMethod
2last digit is even
3Procedure 1: a=1
4has its last 2 digits divisible by 4
5ends in 0 or 5
6is divisible by 2 and 3
7Procedure 2. a=3
8has its last 3 digits divisible by 8
9Procedure 1. a=1
10ends in 0
11Procedure 1. a=2
12is divisible by 3 and 4
13Procedure 2. a=3
14is divisible by 2 and 7
15is divisible by 3 and 5
16has its last 4 digits divisible by 16
17Procedure 2: a=8
18is divisible by 2 and 9
19Procedure 2: a=9
20last digit is zero and tens digit is even
21is divisible by 3 and 7
22is divisible by 2 and 11
23Procedure 2: a=11
24is divisible by 3 and 8
25ends in 00, 25, 50, or 75
26is divisible by 2 and 13
27Procedure 1: a=3
28is divisible by 4 and 7
29Procedure 2: a=14
30is divisible by 3 and 10
31Procedure 1: a=15
32has its last 5 digits divisible by 32
33is divisible by 3 and 11
34is divisible by 2 and 17
35is divisible by 5 and 7
36is divisible by 4 and 9
37Procedure 1: a=3
38is divisible by 2 and 19
39is divisible by 3 and 13
40ends in zero, two digits to left of zero are divisible by 4
41Procedure 1: a=5
42is divisible by 2, 3, and 7
43Procedure 1: a=21
44is divisible by 4 and 11
45is divisible by 5 and 9
46is divisible by 2 and 23
47Procedure 2: a=23
48is divisible by 3 and 16
49Procedure 2: a=21
50ends in 00 or 50
51is divisible by 3 and 17
52is divisible by 4 and 13
53Procedure 1: a=13
54is divisible by 2 and 27
55is divisible by 5 and 11
56is divisible by 7 and 8
57is divisible by 3 and 19
58is divisible by 2 and 29
59Procedure 2: a=29
60is divisible by 3 and 20
61Procedure 2: a=30
62is divisible by 2 and 31
63is divisible by 7 and 9
64has its last 6 digits divisible by 64
65is divisible 5 and 13
66is divisible by 2, 3, and 11
67Procedure 1: a=33
68is divisible by 4 and 17
69is divisible by 3 and 23
70is divisible by 7 and 10
71Procedure 1: a=35
72is divisible by 8 and 9
73Procedure 1: a=4
74is divisible by 2 and 37
75is divisible by 3 and 25
76is divisible by 4 and 19
77is divisible by 7 and 11
78is divisible by 2, 3, and 13
79Procedure 1: a=13
80ends in zero, three digits to left of zero are divisible by 8
81Procedure 1: a=9
82is divisible by 2 and 41
83Procedure 1: a=41
84is divisible by 3, 4, and 7
85is divisible by 5 and 17
86is divisible by 2 and 43
87is divisible by 3 and 29
88is divisible by 8 and 11
89Procedure 1: a=44
90is divisible by 9 and 10
91is divisible by 7 and 13
92is divisible by 4 and 23
93is divisible by 3 and 31
94is divisible by 2 and 47
95is divisible by 5 and 19
96is divisible by 3 and 32
97Procedure 2: a=48
98is divisible by 2 and 49
99Procedure 1: a=2
100ends in 00
101Procedure 2: a=2
102is 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.

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Last Updated: 2009-05-02