I was working with some students on divisibility checks a few days ago, and they asked me if there were any easy checks for numbers larger than 11 (I wrote up a check for every number up to 11 here).

So, here are some more checks, for every number up to 25.

Our general strategy will be as follows:

If the number we're checking divisibility by is composite, say 14, we'll check two different (relatively prime) factors of the number--factors we already know the rules for. In our example 14 = 2 × 7, so we'll check the rules for 2 and 7. The exceptions to this are 16 and 25, which are powers of prime numbers and must be handled differently.

If the number we're checking divisibility by is prime, such as 17, we'll use a "separate and recombine" method similar to the approach we used for 7. We'll separate the last digit of our number to test from the other digits, multiply it by some number, and either add or subtract it from the remaining digits (depending on what number we're checking divisibility by). Follow the link to the earlier tests for divisibility to see how the rule specifically works for 7.

So, without further ado, let's get started.

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12:

If a number is divisible by both 3 and 4, it's divisible by 12.

Example: 168432

1 + 6 + 8 + 4 + 3 + 2 = 24, which is divisible 3, so 168432 is divisible by 3.

32 is divisible by 4, so 168432 is divisible by 4.

Thus 168432 is divisible by 12.

13:

Separate the last digit from the others, multiply that digit by 4, and add it to the remaining digits. Keep doing this until you get a small number whose divisibility you can check directly.

Example: 107393

Separate the last digit from the rest: 10739 | 3

Multiply it by 4 and add that to the remaining digits: 10739 + (4 × 3) = 10739 + 12 = 10751.

Do it again: 1075 + (4 × 1) = 1075 + 4 = 1079

Another round: 107 + (4 × 9) = 107 + 36 = 143

One more time: 14 + (4 × 3) = 14 + 12 = 26

26 is divisible by 13, so 107393 is divisible by 13.

14:

If a number is divisible by both 2 and 7, it's divisible by 14.

Example: 141624

141624 is even, so it's divisible by 2.

14162 - (2 × 4) = 14154, 1415 - (2 × 4) = 1407, 140 - (2 × 7) = 126, 12 - (2 × 6) = 0. 0 is divisible by 7, so 141624 is divisible by 7.

Thus 141624 is divisible by 14.

15:

If a number is divisible by both 3 and 5, it's divisible by 15.

Example: 493815

4 + 9 + 3 + 8 + 1 + 5 = 30, which is divisible by 3, so 493815 is divisible by 3.

493815 ends in 5, so it's divisible by 5.

This 493815 is divisible by 15.

16:

Throw out all the digits except the last four. If the thousands digit is even, throw it out; if it's odd, add 8 to the other three digits and then throw out the thousands digit.

Separate the hundreds digit from the other digits, multiply it by 4, and add it to the other number. If the result is divisible by 16, the original number is divisible by 16.

Example: 949312

Throw out all but the last four digits: 9312

The thousands digit is odd, so add 8: 9320

Throw out the thousands digit: 320

Separate the hundreds digit: 3 | 20

Multiple 3 by 4 and add it to 20: (4 × 3) + 20 = 12 + 20 = 32

32 is divisible by 16, so 949312 is divisible by 16.

17:

Separate the last digit from the others, multiply that digit by 5, and subtract it from the remaining digits. Keep doing this until you get a small number whose divisibility you can check directly.

Example: 534803

Separate the last digit from the rest: 53480 | 3

Multiply it by 5 and subtract that from the remaining digits: 53480 - (5 × 3) = 53480 - 15 = 53465

Do it again: 5346 - (5 × 5) = 5346 - 25 = 5321

Another round: 532 - (5 × 1) = 532 - 5 = 527

One more time: 52 - (5 × 7) = 52 - 35 = 17

17 is divisible by 17, so 534803 is divisible by 17.

18:

If the number is divisible by both 2 and 9, it's divisible by 18.

Example: 114282

114282 is even, so it's divisible by 2.

1 + 1 + 4 + 2 + 8 + 2 = 18, which is divisible by 9, so 114282 is divisible by 9.

Thus 114282 is divisible by 18.

19:

Separate the last digit from the others, multiply that digit by 2, and add it to the remaining digits. Keep doing this until you get a small number whose divisibility you can check directly.

Example: 890701

Separate the last digit from the rest: 89070 | 1

Multiply it by 2 and add that to the remaining digits: 89070 + (2 × 1) = 89070 + 2 = 89072

Do it again: 8907 + (2 × 2) = 8907 + 4 = 8911

Another round: 891 + (2 × 1) = 891 + 2 = 893

One more time: 89 + (2 × 3) = 89 + 6 = 95

95 is divisible by 19, so 890701 is divisible by 19.

20:

If the last two digits are divisible by 20, the original number is divisible by 20.

Example: 792860

60 is divisible by 20, so 792860 is divisible by 20.

21:

If the number is divisible by both 3 and 7, it's divisible by 21.

Example: 202167

2 + 0 + 2 + 1 + 6 + 7 = 18, which is divisible by 3, so 202167 is divisible by 3.

20216 - (2 × 7) = 20202, 2020 - (2 × 2) = 2016, 201 - (2 × 6) = 189, 18 - (2 × 9) = 0. 0 is divisible by 7, so 202167 is divisible by 7.

Thus 202167 is divisible by 21.

22:

If the number is divisible by both 2 and 11, it's divisible by 22.

Example: 492514

492514 is even, so it's divisible by 2.

4 - 9 + 2 -5 + 1 - 4 = -11. -11 is divisible by 11, so 492514 is divisible by 11.

Thus 492514 is divisible by 22.

23:

Separate the last digit from the others, multiply that digit by 7, and add it to the remaining digits. Keep doing this until you get a small number whose divisibility you can check directly.

Example: 839017

Separate the last digit from the rest: 83901 | 7

Multiply it by 7 and add that to the remaining digits: 83901 + (7 × 7) = 83901 + 49 = 83950

Do it again: 8395 + (7 × 0) = 8395 + 0 = 8395

Another round: 839 + (7 × 5) = 839 + 35 = 874

One more time: 87 + (7 × 4) = 87 + 28 = 115

115 is divisible by 23, so 839017 is divisible by 23.

24:

If the number is divisible by both 3 and 8, it's divisible by 24.

Example: 962568

9 + 6 + 2 + 5 + 6 + 8 = 36, which is divisible by 3, so 962568 is divisible by 3.

568 ÷ 2 = 284. 84 is divisible by 4, so 962568 is divisible by 8.

Thus 962568 is divisible by 24.

25:

If the last two digits are divisible by 25, the original number is divisible by 25.

Example: 728175

75 is divisible by 25, so 728175 is divisible by 25.

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If you want some more information on the "separate and recombine" method, this note gives algorithms for every prime number less than 50. And for the theoretically-minded, this page has some information on why the method works.

Have fun, FG.