2 | The last digit is divisible by 2 or is equal to 0. |
3 |
The sum of the digits is divisible by 3.
or
Calculate the difference between double the last digit and the number excluding the last digit.
Repeat until the result is 12 or less.
If the result is a multiple of 3 or is 0 then the number is divisible by 3.
(Observe: 12 => 2 times 2 minus 1 = 3)
or
Calculate the sum of 1 times the last digit and the number excluding the last digit.
Repeat until the result is 9 or less.
If the result is 3, 6, or 9 then the number is divisible by 3.
(Observe: 12 => 1 times 2 plus 1 = 3)
|
4 | The last two digits is divisible by 4 or is equal to 0. |
5 | The last digit is 0 or 5 |
6 | The number is divisible by 2 and 3 |
7 |
Calculate the difference between double the last digit and the number excluding the last digit.
Repeat until the result is less than 10.
If the result is 7 or 0 then the number is divisible by 7.
(Observe: 14 => 4 times 2 minus 1 = 7)
|
8 |
The last three digits is divisible by 8 or is equal to 0. |
9 |
Sum the digits of the number.
If the result is greater than 9 then sum the digits of the result.
Continue until the result is 9 or less.
If the result is 9 then the number is divisible by 9.
or
Calculate the sum of 1 times the last digit and the number excluding the last digit.
Repeat until the result is 9 or less.
If the result is 9 then the number is divisible by 9.
(Observe: 108 => 8 plus 10 = 18 => 1 + 8 = 9)
|
10 |
The last digit is 0. |
11 |
Calculate the difference between the sum of the odd digits and the sum of the even digits.
Repeat the process until the result is 11 or less.
If the result is 11 or 0 then the number is divisible by 11.
or
Calculate the sum of 10 times the last digit and the number excluding the last digit.
Repeat until the result is 11 or less or the same result.
If the result is 11 or the same result then the number is divisible by 11.
(Observe: 11 => 10 times 1 plus 1 = 11)
(Observe: 187 => 10 times 7 plus 18 = 88 => 10 times 8 plus 8 = 88)
(Observe: 209 => 10 times 9 plus 20 = 110 => 10 times 0 plus 11 = 11)
|
12 |
The number is divisible by 4 and 3. |
13 |
Calculate the sum of 4 times the last digit and the number excluding the last digit.
Repeat until the result is 13 or less.
If the result is 13 then the number is divisible by 13.
(Observe: 13 => 1 plus 4 times 3 = 13)
|
14 |
The number is divisible by 2 and 7. |
15 |
The number is divisible by 5 and 3. |
16 |
The last 4 digits is divisible by 16 or is equal to 0. |
17 |
Calculate the difference between 5 times the last digit and the rest of the number.
If the result exceeds 17 then repeat the process.
If the result is 17 or 0 then the number is divisible by 17.
(Observe: 17 => 5 times 7 minus 1 = 34 => 5 times 4 minus 3 = 17)
|
18 |
The number is divisible by 2 and 9. |
19 |
Calculate the sum of 2 times the last digit and the number excluding the last digit.
Repeat until the result is 19 or less.
If the result is 19 or 0 then the number is divisible by 19.
(Observe: 19 => 9 times 2 plus 1 = 19)
|
20 |
Divisible by 2 and 10. |
21 |
Divisible by 3 and 7.
or
Calculate the sum of 19 times the last digit and the number excluding the last digit.
Repeat until the result is 21 or less or the same result.
If the result is 21 or the same result then the number is divisible by 21.
(Observe: 21 => 19 times 1 plus 2 = 21)
(Observe: 357 => 19 times 7 plus 35 = 168 => 19 times 8 plus 16 = 168)
(Observe: 399 => 19 times 9 plus 39 = 210 => 19 times 0 plus 21 = 21)
|
22 |
Divisible by 2 and 11. |
23 |
Calculate the sum of 7 times the last digit and the number excluding the last digit.
Repeat until the result is 23 or less.
If the result is 23 then the number is divisible by 23.
(Observe: 23 => 7 times 3 plus 2 = 23)
|
24 |
Divisible by 4 and 6. |
25 |
The last two digits are 0 , 25 , 50 , or 75. |
26 |
Divisible by 2 and 13. |
27 |
Calculate the difference between 8 times the last digit and the number excluding the last digit.
Repeat until the result is 27 or less.
If the result is 27 or 0 then the number is divisible by 27.
(Observe: 27 => 8 times 7 minus 2 = 54 => 8 times 4 minus 5 = 27)
|
28 |
Divisible by 4 and 7.
or
Divisible by 2 and 14.
|
29 |
Calculate the sum of 3 times the last digit and the number excluding the last digit.
Repeat until the result is 29 or less.
If the result is 29 then the number is divisible by 29.
(Observe: 29 => 3 times 9 plus 2 = 29)
|
31 |
Calculate the sum of 28 times the last digit and the number excluding the last digit.
Repeat until the result is 31 or less or results in the same result.
If the result is 31 or the same result then the number is divisible by 31.
(Observe: 31 => 28 times 1 plus 3 = 31)
(Observe: 775 => 28 times 5 plus 77 = 217 => 28 times 7 plus 21 = 217)
(Observe: 837 => 28 times 7 plus 83 = 279 => 28 times 9 plus 27 = 279)
|
33 |
Calculate the sum of 10 times the last digit and the number excluding the last digit.
Repeat until the result is 33 or less.
If the result is 33 then the number is divisible by 33.
(Observe: 132 => 10 times 2 plus 13 = 33)
|
37 |
Calculate the difference between 8 times the last digit and the number excluding the last digit.
Repeat until the result is 27 or less.
If the result is 27 or 0 then the number is divisible by 27.
(Observe: 37 => 11 times 7 minus 3 = 74 => 11 times 4 minus 7 = 37)
|
41 |
Calculate the sum of 37 times the last digit and the number excluding the last digit.
Repeat until the result is 41 or less or results in the same result.
If the result is 41 or the same result then the number is divisible by 41.
(Observe: 41 => 37 times 1 plus 4 = 41)
(Observe: 943 => 37 times 3 plus 94 = 205 => 37 times 5 plus 20 = 205)
|
99 |
Calculate the sum of 10 times the last digit and the number excluding the last digit.
Repeat until the result is 99 or less.
If the result is 99 then the number is divisible by 99.
(Observe: 1287 => 10 times 7 plus 128 = 198 => 10 times 8 plus 19 => 99)
|
239 |
Calculate the sum of 24 times the last digit and the number excluding the last digit.
Repeat until the result is 239 or less.
If the result is 239 then the number is divisible by 239.
(Observe: 6931 => 24 times 1 plus 693 = 717 => 24 times 7 plus 71 = 239)
|
xxx1 (any divisor ending in 1) |
Calculate the sum of (xxx times 9 plus 1) times the last digit and the number excluding the last digit.
Repeat until the result is xxx1 or less or results in the same result.
If the result is xxx1 or the same result then the number is divisible by xxx1.
(Observe: 41 => 37 times 1 plus 4 = 41)
(Observe: 943 => 37 times 3 plus 94 = 205 => 37 times 5 plus 20 = 205)
|
xxx3 (any divisor ending in 3) |
Calculate the sum of (xxx times 3 plus 1) times the last digit and the number excluding the last digit.
Repeat until the result is xxx3 or less.
If the result is xxx3 then the number is divisible by xxx3.
Note: For the case of 3 the result is 3 , 6 , or 9.
(Observe: 132 => 10 times 2 plus 13 = 33 ; times 4 = 132)
|
xxx7 (any divisor ending in 7) |
Calculate the difference between (xxx times 3 plus 2) times the last digit and the number excluding the last digit.
Repeat until the result is xxx7 or less.
If the result is xxx7 or 0 then the number is divisible by xxx7.
(Observe: 37 => 11 times 7 minus 3 = 74 => 11 times 4 minus 7 = 37)
|
xxx9 (any divisor ending in 9) |
Calculate the sum of (xxx + 1) times the last digit and the number excluding the last digit.
Repeat until the result is xxx9 or less.
If the result is xxx9 then the number is divisible by xxx9.
|