# Divisibility Rules

Knowing how an integer can be divided is useful when simplifying fractions, distributing items equally among people, or performing long division by hand. There are simple divisibility tests for the prime numbers 2, 3, 5, and 11, plus several of the composite numbers formed by these primes.

There are also divisibility tests for other integers including 7, 13, etc., though these techniques are more cumbersome to apply. (In some cases, you might find it easier to do long division instead.) Here is the list of the rules for checking divisibility for the numbers 2 through 41.

**Division by 2:**

If the number ends in 0, 2, 4, 6, or 8, then the number is divisible by 2. This is equivalent to being an even number.

**Division by 3:**

Add up all of the digits in the number. If the sum of the digits is divisible by 3, then so is the number.

**Division by 4:**

If the last two digits of the number is divisible by 4 then so is the number. For example, the number 93548 is divisible by 4 since 48 is divisible by 4.

**Division by 5:**

If the last digit is 0 or 5 then the number can be divided by 5.

**Division by 6:**

Apply the rule for 2 and 3. If the number passes both tests, then the number is divisible by 6.

**Division by 7--First Method:**

Double the last digit and subtract it from the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 7, then so is the larger number. If this smaller number is not divisible by 7, then neither is the larger number. For example, let's check the divisibility of 864503:

86450**3**, 3x2 = 6 and 86450-6 = 86444

8644**4**, 4x2 = 8 and 8644-8 = 8636

863**6**, 6x2 = 12 and 863-12 = 851

85**1**, 1x2 = 2 and 85-2 = 83

Since 83 is not divisible by 7, neither is the original number 864503.

**Division by 7--Second Method:**

Multiply the last digit by 5 and add it to the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 7, then so is the larger number. If this smaller number is not divisible by 7, then neither is the larger number. For instance, suppose we check the divisibility of 864503 using this technique:

86450**3**, 3x5 = 15 and 86450+15 = 86465

8646**5**, 5x5 = 25 and 8646+25 = 8671

867**1**, 1x5 = 5 and 867+5 = 872

87**2**, 2x5 = 10 and 87+10 = 97

Since 97 is not divisible by 7, neither is the original number 864503.

**Division by 7--Third Method:**

This technique works for large numbers: Take the last three digits of the number and subtract this from the number formed by the remaining digits. Repeat this process until you end up with a number that has at most three digits. At that point you may apply either the first method or the second method. For example, let's check if 3718549877 is divisible by 7:

3718549**877**, 3718549-877 = 3717672

3717**672**, 3717-672 = 3045

3**045**, 3-45 = -42

Since -42 is divisible by 7, then 3718549877 is also divisible by 7.

**Division by 8:**

Check the last three digits of the number. If it forms an integer that is divisible by 8, then the number is also divisible by 8. For instance, 73540665742 is not divisible by 8 since 742 is not divisible by 8.

**Division by 9:**

Add up all of the digits in the number. If the sum of the digits is divisible by 9, then so is the number.

**Division by 10:**

If the last digit is 0, then the number is divisible by 10.

**Division by 11:**

Alternately add and subtract all of the digits of the number, starting with subtraction on the second digit. If the result is 0 or any number divisible by 11, then so is the number. For example, consider the number 119637360799. If we compute

1-1+9-6+3-7+3-6+0-7+9-9

we get a total of -11. Since -11 is divisible by 11, so is 119637360799.

**Division by 12:**

Apply the rules for 3 and 4. If the number passes both tests, then the number is divisible by 12.

**Division by 13--First Method:**

Multiply the last digit by 9 and subtract it from the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 13, then so is the larger number. If this smaller number is not divisible by 13, then neither is the larger number. For example, let's check the divisibility of 399074:

39907**4**, 4x9 = 36 and 39907-36 = 39871

3987**1**, 1x9 = 9 and 3987-9 = 3978

397**8**, 8x9 = 72 and 397-72 = 325

32**5**, 5x9 = 45 and 32-45 = -13

Since -13 is divisible by 13, then 399074 is also divisible by 13.

**Division by 13--Second Method:**

Multiply the last digit by 4 and add it to the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 13, then so is the larger number. If this smaller number is not divisible by 13, then neither is the larger number. For example, let's check the divisibility of 399074:

39907**4**, 4x4 = 16 and 39907+16 = 39923

3992**3**, 3x4 = 12 and 3992+12 = 4004

400**4**, 4x4 = 16 and 400+16 = 416

41**6**, 6x4 = 24 and 41+24 = 65

Since 65 is divisible by 13, then 399074 is also divisible by 13.

**Division by 13--Third Method:**

This is the same technique as for 7: Take the last three digits of the number and subtract this from the number formed by the remaining digits. Repeat this process until you end up with a number that has at most three digits. At that point you may apply either the first method or the second method to check if the number is divisible by 13. For example, let's check if 16049371 is divisible by 13:

16049**371**, 1234-580 = 15678

15**678**, 15-678 = -663

Now use the second method to check if 663 is divisible by 13:

66**3**, 3x4 = 12 and 66+12 = 78

7**8**, 8x4 = 32 and 7-32 = -26

Since -26 is divisible by 13, then 16049371 is also divisible by 13.

**Division by 14:**

Apply the rule for 2 and one of the rules for 7. If the number passes both divisibility tests, then the number can be divided by 14.

**Division by 15:**

Apply the rules for 3 and 5. If the number passes both tests, then the number is divisible by 15.

**Division by 16:**

Check the last 4 digits of the number. If the last 4 digits form an integer that is divisible by 16, then the original number is also divisible by 16. For instance, 157675552 can be divided by 16 since 5552 is a multiple of 16.

**Division by 17:**

Multiply the last digit by 5 and subtract it from the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 17, then so is the larger number. If this smaller number is not divisible by 17, then neither is the larger number. For example, let's check the divisibility of 521172:

52117**2**, 2x5 = 10 and 52117-10 = 52107

5210**7**, 7x5 = 35 and 5210-35 = 5175

517**5**, 5x5 = 25 and 517-25 = 492

49**2**, 2x5 = 10 and 49-10 = 39

Since 39 is not divisible by 17, then neither is 521172.

**Division by 18:**

Apply the rules for 2 and 9. If the number passes both tests, it is divisible by 18.

**Division by 19:**

Multiply the last digit by 2 and add it to the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 19, then so is the larger number. If this smaller number is not divisible by 19, then neither is the larger number. For example, let's check the divisibility of 12483:

1248**3**, 3x2 = 6 and 1248+6 = 1254

125**4**, 4x2 = 8 and 125+8 = 133

13**3**, 3x2 = 6 and 13+6 = 19

Since 19 is divisible by 19, then so is 12483.

**Division by 20:**

Apply the rules for 4 and 5. If the number passes both tests, it is divisible by 20.

**Division by 21:**

Apply the rule for 3 and one of the rules for 7. If the number passes both tests, it is divisible by 21.

**Division by 22:**

Apply the divisibility tests for 2 and 11. If the number meets both conditions, it is divisible by 22.

**Division by 23:**

Multiply the last digit by 7 and add it to the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 23, then so is the larger number. If this smaller number is not divisible by 23, then neither is the larger number. For example, let's check the divisibility of 53682:

5368**2**, 2x7 = 14 and 5368+14 = 5382

538**2**, 2x7 = 14 and 538+14 = 552

55**2**, 2x7 = 14 and 55+14 = 69

6**9**, 9x7 = 63 and 6+63 = 69

Since 69 is divisible by 23, 53682 is also divisible by 23.

**Division by 24:**

Apply the tests for 3 and 8. If the number passes both tests, then the number is a multiple of 24.

**Division by 25:**

If the last two digits are 00, 25, 50, or 75, then the number can be divided by 25.

**Division by 26:**

Apply the rule for 2 and one of the rules for 13. If the number passes both tests, it is divisible by 26.

**Division by 27:**

Multiply the last digit by 8 and subtract it from the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 27, then so is the larger number. If this smaller number is not divisible by 27, then neither is the larger number. For example, let's check the divisibility of 10962:

1096**2**, 2x8 = 16 and 1096-16 = 1080

108**0**, 0x8 = 0 and 108-0 = 108

10**8**, 8x8 = 64 and 10-64 = -54

Since -54 is divisible by 27, 10962 is also divisible by 27.

**Division by 28:**

Apply the rule for 4 and one of the rules for 7. If the number passes both tests, it is divisible by 28.

**Division by 29:**

Multiply the last digit by 3 and add it to the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 29, then so is the larger number. If this smaller number is not divisible by 29, then neither is the larger number. For example, let's check the divisibility of 24273:

2427**3**, 3x3 = 9 and 2427+9 = 2436

243**6**, 6x3 = 18 and 243+18 = 261

26**1**, 1x3 = 3 and 26+3 = 29

Since 29 is divisible by 29, then 24273 is as well.

**Division by 30:**

Apply the rules for 2, 3, and 5. If the number passes all three tests, then it is divisible by 30.

**Division by 31:**

Multiply the last digit by 3 and subtract it from the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 31, then so is the larger number. If this smaller number is not divisible by 31, then neither is the larger number. For example, let's check the divisibility of 504273:

50427**3**, 3x3 = 9 and 50427-9 = 50418

5041**8**, 8x3 = 24 and 5041-24 = 5017

501**7**, 7x3 = 21 and 501-21 = 480

48**0**, 0x3 = 0 and 48-0 = 48

Since 48 is not divisible by 31, then neither is 504273.

**Division by 32:**

Check the last 5 digits of the number. If the last 5 digits form an integer that is divisible by 32, then the original number is also divisible by 32. For instance, 31999968 is divisible by 32 since 99968 is a multiple of 32.

**Division by 33:**

Apply the rule for 3 and one of the rules for 11. If the number passes both tests, it is divisible by 33.

**Division by 34:**

Apply the rule for 2 and one of the rules for 17. If the number passes both tests, it is divisible by 34.

**Division by 35:**

Apply the rules for 5 and 7. If the number passes both tests, it is divisible by 35.

**Division by 36:**

Apply the rules for 4 and 9. If the number passes both tests, it is divisible by 36.

**Division by 37:**

Take the last three digits of the number and add this to the number formed by the remaining digits. Repeat this process until you end up with a number that has at most three digits. If the remaining number is divisible by 37, then so is the larger number. For example, let's test the divisibility of 361975218:

361975**218**, 361975+218 = 362193

362**193**, 362+193 = 555

Since 555 is divisible by 37, then 361975218 is also divisible by 37.

**Division by 38:**

Apply the rules for 2 and 19. If the number passes both tests, it is divisible by 38.

**Division by 39:**

Apply the rule for 3 and one of the rules for 13. If the number passes both tests, it is divisible by 39.

**Division by 40:**

Apply the rules for 5 and 8. If the number passes both tests, it is divisible by 40.

**Division by 41:**

Multiply the last digit by 4 and subtract it from the number formed by the remaining digits. Repeat this process until you arrive at a smaller number whose divisibility you know. If this smaller number is divisible by 41, then so is the larger number. If this smaller number is not divisible by 41, then neither is the larger number. For example, let's check the divisibility of 142311:

14231**1**, 1x4 = 4 and 14231-4 = 14227

1422**7**, 7x4 = 28 and 1422-28 = 1394

139**4**, 4x4 = 16 and 139-16 = 123

12**3**, 3x4 = 12 and 12-12 = 0

Since 0 is divisible by 41, then so is 142311.

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