What are the divisibility rules for 2 through 12?

Answer 1

To be divisible by 2 the last digit must be even, ie one of {0, 2, 4, 6, 8};

For example 1234567: The last digit is 7, which is not one of these so it is not divisible by 2.

To be divisible by 3, sum the digits of the number and if this sum is divisible by 3, then the original number is divisible by 3.

As the test can be repeated on the sum, repeat the summing until a single digit remains; only if this number is one of {3, 6, 9} is the original number divisible by 3.

For example 1234567:

1234567 → 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

28 → 2 + 8 = 10

10 → 1 + 0 = 1

1 is not one of {3, 6,9} so 1234567 is not divisible by 3.

To be divisible by 4, add twice the tens digit to the ones digit; if this sum is divisible by 4, then so is the original number.

As the test can be repeated on the sum, repeat the summing until a single digit remains; only if this number is one of {4, 8} is the original number divisible by 4.

For example 4562364:

→ 2×6 + 4 = 16

→ 2×1 + 6 = 8

8 is one of {4, 8} so 4562364 is divisible by 4.

To be divisible by 5, the last digit must be one of {0, 5}.

For example: 48250

The last digit is 0 which is not one of {0, 5} so it is divisible by 5.

To be divisible by 6, the number must be divisible by both 2 and 3; apply the tests for both of these.

for example: 346734

Last digit is 4, which is even (one of {0, 2, 4, 6, 8}) so 346734 is divisible by 2

346734 → 3 + 4 + 6 + 7 + 3 + 4 = 27

27 → 2 + 7 = 9

9 is one of {3, 6, 9}, so 346734 is divisible by 3

346734 is divisible by both 2 & 3 → 346734 is divisible by 6

There is no real check for 7 which is not much slower than just dividing by 7 to see if there is no remainder. One check:

Write the digits in blocks of 3 starting from the right hand end (like you would for reading the number):

in each block of 3 add twice the first digit to three times the second digit to the third digit.

Alternately subtract and add the blocks starting from the right hand end of the number.

If the result is divisible by 7, then so is the original number.

For example: 15,838,874

15,838,874 → 2×0 + 3×1 + 5, 2×8 + 3×3 + 8, 2×8 + 3×7 + 4

→ 8, 33, 41

→ 41 - 33 + 8 = 16

16 is not divisible by 7, so 15,838,874 is not divisible by 7.

To be divisible by 8, add four times the hundreds digit to twice the tens digit to the ones digit; if this sum is divisible by 8, then so is the original number. As the test can be repeated on the sum, repeat the summing until a single digit remains; only is this number is 8 is the original number divisible by 8.

For example: 34689312572

→4×5 + 2×7 + 2 = 36

36 → 4×0 + 2×3 + 6 = 12

12 → 4×0 + 2×1 + 2 = 4

4 is not 8 so 34689312572 is not divisible by 8.

To be divisible by 9, sum the digits of the number and if this sum is divisible by 9, then the original number is divisible by 9.

As the test can be repeated on the sum, repeat the summing until a single digit remains (this single digit is called the digital root of the number); only if this number is 9 is the original number divisible by 9.

For example 85423479674

85423479674 → 8 + 5 + 4 + 2 + 3 + 4 + 7 + 9 + 6 + 7 + 4 = 59

59 → 5 + 9 = 14

14 → 1 + 4 = 5

5 is not 9 so 85423479674 is not divisible by 9.

To be divisible by 10, the last digit must be 0

For example: 346759056; the last digit is 6 which is not 0, so it is not divisible by 10.

To be divisible by 11, alternately subtract and add the digits of the number from the right hand end; only if this sum is divisible by 11 (or is 0) is the original number divisible by 11.

For example: 4690263597

4690263597 → 7 - 9 + 5 - 3 + 6 - 2 + 0 - 9 + 6 - 4 = -3

-3 is not divisible by 11, so 4690263597 is not divisible by 11.

To be divisible by 12, the the number must be divisible by both 3 and 4; apply the tests for both of these.

for example: 3467343564

3467343564 → 3 + 4 + 6 + 7 + 3 + 4 + 3 + 5 + 6 + 4 = 45

45 → 4 + 5 = 9

9 is one of {3, 6, 9} so 3467343564 is divisible by 3

64 → 2×6 + 4 = 16

16→ 2×1 + 6 = 8

8 is one of {4, 8} so 3467343564is divisible by 4

→ 3467343564is divisible by both 3 and 4, so 3467343564 is divisible by 12