Divisibility Tests

Dividing by 2
If the number ends in a even digit, it is divisible by 2
Example:
22 is divisible by 2
15421464605613404646518 is divisible by 2



Dividing by 3
Add up the digits: if the sum is divisible by three, then the number is as well. Examples:
111111: the digits add to 6 so the whole number is divisible by three.
87687687. The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by 3



Dividing by 4
Look at the last two digits. If the number formed by its last two digits is divisible by 4, the original number is as well.
Examples:
100 is divisible by 4.
1732782989264864826421834612 is divisible by four also, because 12 is divisible by four.



Dividing by 5
If the last digit is a five or a zero, then the number is divisible by 5.
Examples:
125 is divisible by 5
175983421545613651120 is divisible by 5



Dividing by 6
Check 3 and 2. If the number is divisible by both 3 and 2, it is divisible by 6 as well.



Dividing by 7
Use the 3 coefficients (1 , 2 , 3). Multiply the first number starting from the ones place by 1, then the second from the right by 3, the third by 2, the fourth by -1, the fifth by -3, the sixth by -2, and the seventh by 1, and so forth.

Example: 348967129356876.

6 + 21 + 16 - 6 - 15 - 6 + 9 + 6 + 2 - 7 - 18 - 18 + 8 + 12 + 6 = 16
means the number is not multiple of seven.

If the number was 348967129356874, then the number is a multiple of seven
because instead of 16, we would find 14 as a result, which is a multiple of 7.

So the pattern is as follows: for a number onmlkjihgfedcba, calculate

a + 3b + 2c - d - 3e - 2f + g + 3h + 2i - j - 3k - 2l + m + 3n + 2o.

Example: 348967129356874.

Below each digit let me write its respective figure.

3 4 8 9 6 7 1 2 9 3 5 6 8 7 6
2 3 1 -2 -3 -1 2 3 1 -2 -3 -1 2 3 1

(3×2) + (4×3) + (8×1) + (9×-2) + (6×-3) + (7×-1) +
(1×2) + (2×3) + (9×1) + (3×-2) + (5×-3) + (6×-1) +
(8×2) + (7×3) + (6×1) = 16 -- not a multiple of 7.



Dividing by 8
Check the last three digits. Since 1000 is divisible by 8, if the last three digits of a number are divisible by 8, then so is the whole number.
Example: 33333888 is divisible by 8; 33333886 isn't.



Dividing by 9
Add the digits. If that sum is divisible by nine, then the original number is as well.



Dividing by 10
If the number ends in 0, it is divisible by 10.



Dividing by 11
Let's look at 352, which is divisible by 11; the answer is 32. 3+2 is 5; another way to say this is that 35 -2 is 33.
Now look at 3531, which is also divisible by 11. It is not a coincidence that 353-1 is 352 and 11 × 321 is 3531.

Here is a generalization of this system. Let's look at the number 94186565.

First we want to find whether it is divisible by 11, but on the way we are going to save the numbers that we use: in every step we will subtract the last digit from the other digits, then save the subtracted amount in order. Start with

9418656 - 5 = 9418651 SAVE 5
Then 941865 - 1 = 941864 SAVE 1
Then 94186 - 4 = 94182 SAVE 4
Then 9418 - 2 = 9416 SAVE 2
Then 941 - 6 = 935 SAVE 6
Then 93 - 5 = 88 SAVE 5
Then 8 - 8 = 0 SAVE 8
Now write the numbers we saved in reverse order, and we have 8562415, which multiplied by 11 is 94186565.



Dividing by 12
Check for divisibility by 3 and 4.



Dividing by 13
Delete the last digit from the given number. Then subtract nine times the deleted digit from the remaining number. If what is left is divisible by 13, then so is the original number.