- For explanation of general divisibility rules explained, see Divisibility Rules.
Below is a List of Divisibility Rules sorted by number.
Difficulty CodingEdit
Very Easy | Easy | Somewhat Easy | Medium | Moderately Hard | Difficult | Very Difficult | Totally Nuts |
Additionally, if an alternative method is available (albeit it would be hard most of the time), the description for that number has a pink background.
Divisibility of Numbers below 5Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
1 | Every integer is divisible by one. All prime numbers are divisible by one, too. (Proof) | 4,623 is divisible by one; 91,237 is divisible by one. | |
2 | Every number including 0 which ends with 0, 2, 4, 6 or 8 is divisible by two. (Proof) | 65,156,151,594 is divisible by two because the units digit is 4, and 1,597,534,568,852 is divisible by two because the units digit is 2. | |
3 | A number is divisible by three if the sum of its digits is divisible by three. (Proof) | Is 12423 divisible by 3? (1) Add all the digits together, we get 1 + 2 + 4 + 2 + 3 = 12. (2) 12 is divisible by 3, so 12423 is, too. | |
4 | A number is divisible by four if and only if its last two digits of the number are divisible by four. (Proof) | 156,128 is divisible by four, and 6,416 is divisible by four. The last two digits are 28 and 16, respectively, both of which are divisible by 4. | |
5 | A number is divisible by five if and only if the last digit is a 0 or 5. (Proof) | 213,478,765 is divisible by 5 because the last digit is 5. |
Divisibility of Numbers between 6 and 10Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
6 | Because its prime factorization is 2 x 3, it should be divisible by those two numbers. | Is 4,044 divisible by 6? 4,044 is even, so it is divisible by 2. 4 + 4 + 4 is 12, so it is divisible by 3. Therefore, 4,044 is indeed a multiple of 6. | |
7 | Take the last digit of the number, multiply it by two, and then subtract it to the remaining digits of the number. Repeat the process until the result can be easily identified. (Proof) Another strategy is to add 5 times the last digit to the remaining digits instead. If the number is too large for the previously mentioned method to be viable then a block strategy can be used. First solit the number in blocks of 3 digits each, then start by substrating the first block with the second, add the third block to the result, substract the fourth one and repeat the pattenr, alternating betweem addition and substraction, until you get one single number. If that result is divisible by 7 the original number is also divisible by 7. | Is 3,409 is divisible by 7? First, take 9 from 3409, and multiply 9 by 2 (9*2=18). Then, subtract the doubled digit to the remaining digits. (340-18=322) Repeat the process: take 2 from 322 and multiply it by 2 (2*2=4). Subtract the doubled digit to the remaining number (32-4=28) 28 is divisible by 7, so 3,409 is also divisible by 7. | |
8 | It is very simple, check whether the last three digits of the number is divisible by 8, if it is, then the number is divisible by 8. | Is the number 7,377,473,496 divisible by 8? First, Take the last three digits of the number. The last three digit is 496, so check whether it is divisible by 8. 496 is divisible by 8. Therefore, 7,377,473,496 is divisible by 8. | |
9 | Add all the digits together. Then, instead of checking divisibility of 3, check if it is the multiple of 9. (Proof) | Is 14625 divisible by 9? Add all the digits together: 1 + 4 + 6 + 2 + 5 = 18 18 is divisible by 9, so 14625 is, too. | |
10 | If the last digit of the number is 0, it is divisible by 10. | 158,375,209,580 is divisible by 10 because the last digit is 0. |
Divisibility of Numbers between 11 and 20Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
11 | If the difference of sum of digits at odd places and sum of digits at even places is 0 or divisible by 11, then the number is divisible by 11. | Is 527,901 divisible by 11? First, Add the digits at odd places. (5+7+0=12) Add the digits at even places. (2+9+1=12) Subtract both sums. (12-12=0) The result is 0, so 527,901 is divisible by 11. | |
12 | Because its prime factorization is 2^{2} x 3, the number needs to be divisible by 4 and 3. | 9,180 ends in 80 (4*20), so it is divisible by 4. The sum of all digits is 18, so it is divisible by 3. Therefore, 9,180 is a multiple of 12. | |
13 | Take the last digit of the number, multiply it by four, and then add it to the remaining digits of the numbers. Repeat the process until the result can be easily identified. Alternatively, subtracting 9 times the last digit to the remaining digits is also a viable strategy. | Take 85,969 for example. Take out 9 from 85969 and multiply it by four. (9 x 4 = 36) Add it to the remaining digits. (8596 + 36 = 8632) Take out 2 from 8632 and multiply it by four. (2 x 4 = 8) Add it to the remaining digits. (863 + 8 = 871) Take out 1 from 871 and multiply it by four. (1 x 4 = 4) Add it to the remaining digits. (87 + 4 = 91) 91 is divisible by 13, so 85,969 is also divisible by 13. | |
14 | As 14 is the product of 2 and 7, perform divisibility checks of both 2 and 7. | For the number 745,514: It is divisible by 2, since it is an even number. Perform divisibility check of 7.Divisibility of 7: Take away the last digit of the number, multiply it by 2, and subtract it from the remaining digits. Repeat the process until the result can be determined. Example: For the number 745,514, 1. Take the last digit of the number and multiply it by 2. (4 x 2 = 8) 2. Subtract that number from the rest of the digits. (74551 - 8 = 74543) 3. Repeat the process. (7454 - 3 x 2 = 7448) 4. Continue until the results can be determined. (744 - 8 x 2 = 728) 5. The result is should be divisible by 7. (72 - 8 x 2 = 56) 56 is divisible by 7. Therefore, 745,514 is divisible by 14. | |
15 | Check if the number is divisible by 3 and 5. | Take 2,955, for example, the sum of all digits is 21, and the last digit of the number is 5. Therefore, 2,955 is divisible by 15. | |
16 | Similar to divisibility by 8, except check if the last 4 digits are divisible by 16 rather than the last 3. | 2,295,632 is divisible by 16, because the last 4 digits is 5,632. | |
17 | Take the last digit of the number, multiply it by five, and then subtract it to the remaining digits of the numbers. Repeat the process until the result can be easily identified. | Take 59,789 for example. Take out 9 from 59789 and multiply it by five. (9 x 5 = 45) Subtract it from the remaining digits. (5978 - 45 = 5933) Continue repeating the same steps. (593 - 3 x 5 = 578; 57 - 8 x 5 = 17) 17 is divisible by 17, so 59,789 is also divisible by 17. | |
18 | Because its prime factorization is 2 x 3^{2}, the number needs to be divisible by 2 and 9. | 671,634 is an even number, and the sum of all digits is 27. Therefore, 671,634 is divisible by 18. | |
19 | Take the last digit of the number, multiply it by two, and then add it to the remaining digits of the numbers. Repeat the process until the result can be easily identified. | Take 3,971 for example. Take out 1 from 3971 and multiply it by two. (1 x 2 = 2) Add it to the remaining digits. (397 + 2 = 399) Continue repeating the same steps. (39 + 9 x 2 = 57) 57 is divisible by 19, so 3,971 is also divisible by 19. | |
20 | Check the last 2 digits: They should end in 20, 40, 60, 80, or 00. | 46,186,940 is divisible by 20 because the last 2 digits is divisible by 20. |
Divisibility of Numbers between 21 and 30Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
21 | Check the divisibility of 3 and 7. | Take 73,857, for example: the sum of digits is 30, making it divisible by 3. Then, perform divisibility check of 7.Divisibility of 7: Take away the last digit of the number, multiply it by 2, and subtract it from the remaining digits. Repeat the process until the result can be determined. Example: For the number 73,857, 1. Take the last digit of the number and multiply it by 2. (7 x 2 = 14) 2. Subtract that number from the rest of the digits. (7385 - 14 = 7371) 3. Repeat the process. (737 - 1 x 2 = 735) 4. Continue until the results can be determined. (73 - 5 x 2 = 63) 5. The result is should be divisible by 7. (6 - 3 x 2 = 0) As 73,857 is divisible by both 3 and 7, 73,857 is divisible by 21. | |
22 | Check the divisibility of 2 and 11. | 33,242 is an even number divisible by 11Divisibility of 11: Sum up the odd digits, then the even digits. Subtract them. Then check if the number is a multiple of 11. Example: For the number 33,242, the sum of all odd digits is 7, and the sum of all even digits is 7. After subtracting them, the result is 0, which is divisible by 11; therefore, 33,242 is also divisible by 11., so 33,242 is divisible by 22. | |
23 | Take the last digit of the number, multiply it by seven, and then add it to the remaining digits of the numbers. Repeat the process until the result can be easily identified. | Take 39,077 for example. Take out 1 from 39077 and multiply it by seven. (1 x 7 = 7) Add it to the remaining digits. (3907 + 49 = 3956) Repeat the same steps. (395 + 6x7 = 437; 43 + 7x7 = 92) 92 is divisible by 23, so 39,077 is also divisible by 23. | |
24 | Check the divisibility of 3 and 8. | 176,376 ends in 376 (divisible by 8), and the digit sum is 30 (divisible by 3). Therefore, 176,376 is divisible by 24. | |
25 | The last 2 digits ends in 25, 50, 75, or 00. | 149,346,775 is divisible by 25. | |
26 | Check the divisibility of 2 and 13. | For the number 2,216,734: It is divisible by 2, since it is an even number. Perform divisibility check of 13.Divisibility of 13: Similar to divisibility of 7, but this time, take away the last digit of the number, multiply it by 4, and add it to the remaining digits. Then repeat the process. Example: For the number 2,216,734, 1. Take the last digit of the number and multiply it by 4. (4 x 4 = 16) 2. Add that number to the rest of the digits. (221673 + 16 = 221689) 3. Repeat the process. (22168 + 9 x 4 = 22204) 4. Continue until the results can be determined. (2220 + 4 x 4 = 2236) 5. The result should be divisible by 13. (223 + 6 x 4 = 247) 6. (24 + 7 x 4 = 52) Alternatively, the alternating sum method can be used instead. First, group the numbers of three from right to left. Then alternate between subtraction and addition. Then apply the above method to determine its divisibility of 13. Example: For number 107,359,993, 1. The alternating sum is 993 - 359 + 107 = 741 2. Examine the result. Use the 1st method again to reduce the digits if needed. 52 is divisible by 13. Therefore, 2,216,734 is divisible by 26. | |
27 | Add up the digits in blocks of three and check if it is divisible by 27. If the result is not easy to determine, subtract 8 times the last digit from the rest of the digits. | Take 9,782,673,561, for example: Sum up the digits in blocks of three. (561 + 673 + 782 + 9 = 2025) Take out the last digit and multiply it by 8. (5 x 8 = 40) Subtract it to the rest of the digits (202 - 40 = 162) Repeat the last two steps. (16 - 2 x 8 = 0) Since 0 is divisible by 27, 9,782,673,561 is also divisible by 27. | |
28 | Check the divisibility of 4 and 7. | For the number 457,772: It is divisible by 4, since 72, the last two digits, is divisible by 4. Perform divisibility check of 7.Divisibility of 7: Take away the last digit of the number, multiply it by 2, and subtract it from the remaining digits. Repeat the process until the result can be determined. Example: For the number 457,772, 1. Take the last digit of the number and multiply it by 2. (2 x 2 = 4) 2. Subtract that number from the rest of the digits. (45777 - 4 = 45773) 3. Repeat the process. (4577 - 3 x 2 = 4571) 4. Continue until the results can be determined. (457 - 1 x 2 = 455) 5. The result is should be divisible by 7. (45 - 5 x 2 = 35) 35 is divisible by 7. Therefore, 457,772 is divisible by 28. | |
29 | To check if a number is divisible by 29, take off the last digit, triple it, and then add it to the rest of the digits. | Take 105,299, for example: Take out 9 from 105,299 and multiply it by three. (9 x 3 = 27) Add it from the remaining digits. (10529 + 27 = 10556) Repeat the same steps. (1055 + 6x3 = 1073; 107 + 3x3 = 116) 116 is divisible by 29, so 105,299 is also divisible by 29. | |
30 | Simply check the divisibility of 3 and 10. | 8146920 is divisible by 30 because the last digit is 0, and the sum of digits is 30. |
Divisibility of Numbers between 31 and 40Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
31 | To check if a number is divisible by 31, take off the last digit, triple it, and then subtract it to the rest of the digits. | For the number 31,279, take out 9, triple it (27), and then subtract it from 3127 (3127-27=3100). 3,100 is divisible by 31, therefore 31,279 is also divisible by 31. | |
32 | Check the last 5 digits, although if the ten thousands digit is an odd number, add 16 to the last 4 digits and check them; if the tens thousand digit is even, directly check the last 4 digits. | For the number 1,913,248, the ten thousands digit is 1; therefore, add 16 to the last 4 digits, 3,248. Afterwards, the result 3264 is divisible by 32, so 1,913,248 is divisible by 32. | |
33 | Add up the digits in groups of two from right to left, and check its divisibility by 33. Or, check the divisibility of 3 and 11 separately. | 12,894,387 is divisible by 33 because the digit sum in groups of two is 87 + 43 + 89 + 12 = 231, divisible by 33. | |
34 | Check its divisibility to both 2 and 17. | ||
35 | Check its divisibility to both 5 and 7. | ||
36 | Check its divisibility to both 4 and 9. | ||
37 | Add up the digits in blocks of three and check if it is divisible by 37. If the result is not easy to determine, subtract 11 times the last digit from the rest of the digits. | Take 13,966,427, for example: Sum up the digits in blocks of three. (427 + 966 + 13 = 1406) Take out the last digit and multiply it by 11. (6 x 11 = 66) Subtract it to the rest of the digits (140 - 66 = 74) Since 74 is divisible by 37, 13,966,427 is also divisible by 37. | |
38 | Check its divisibility to both 2 and 19. | ||
39 | Take out the last digit, multiply it by 4, and then add it to the rest of the digits. | Take 140,673, for example: Take out 3 from 140,673 and multiply it by four. (3 x 4 = 12) Add it from the remaining digits. (14067 + 12 = 14079) Repeat the same steps. (1407 + 9x4 = 1443; 144 + 3x4 = 156) 156 is divisible by 39, so 140,673 is also divisible by 39. | |
40 | Check the last 3 digits and determine its divisibility by 40. |
Divisibility of Numbers between 41 and 50Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
41 | Take out the last digit, multiply it by 4, and then subtract it from the remaining digits. | For the number 661,289, take out 9, quadruple it (36), and then subtract it from 66128 (66128-36=66092). Repeat the steps (6609-2x4=6601; 660-1x4=656; 65-6x4=41). 41 is divisible by 41, therefore 661,289 is also divisible by 41. | |
42 | Check the divisibility of 2, 3, and 7. | ||
43 | Take out the last digit, multiply it by 30, and then subtract it to the remaining digits. | For the number 549,067: 54906 - 7x30 = 54696; 5469 - 6x30 = 5289; 528 - 9x30 = 258; 258 is divisible by 43, and therefore 549,067 is divisible by 43. | |
44 | Check the divisibility of 4 and 11. | ||
45 | Check the divisibility of 5 and 9. | ||
46 | Check the divisibility of 2 and 23. | ||
47 | Take out the last digit, multiply it by 14, and then subtract it from the remaining digits. | ||
48 | Check the divisibility of 3 and 16. | ||
49 | Take out the last digit, multiply it by 5, and then add it to the remaining digits. | ||
50 | Check the last 2 digits. It should end in 50 or 00. |
Divisibility of Numbers between 51 and 60Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
51 | Take out the last digit, multiply it by 5, and then subtract it from the remaining digits. | ||
52 | Check the divisibility of 4 and 13. | ||
53 | Take out the last digit, multiply it by 16, and then add it to the remaining digits. | ||
54 | Check the divisibility of 2 and 27. | ||
55 | Check the divisibility of 5 and 11. | ||
56 | Check the divisibility of 7 and 8. | ||
57 | Check the divisibility of 3 and 19. | ||
58 | Check the divisibility of 2 and 29. | ||
59 | Take out the last digit, multiply it by 6, and then add it to the remaining digits. | ||
60 | Perform divisibility checks for 3 and 20. |
Divisibility of Numbers between 61 and 70Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
61 | Take out the last digit, multiply it by 6, and then subtract it from the remaining digits. | ||
62 | Check the divisibility of 3 and 31. | ||
63 | Check the divisibility of 7 and 9. | ||
64 | Check the last 6 digits. | ||
65 | Perform divisibility checks for 5 and 13. | ||
66 | Perform divisibility checks for 6 and 11. | ||
67 | Take out the last digit, multiply it by 20, and then subtract it from the remaining digits. | ||
68 | Perform divisibility checks for 4 and 17. | ||
69 | Take out the last digit, multiply it by 7, and then add it to the remaining digits. It is unnecessary to check divisibility of 3 and 23 individually. | ||
70 | Perform divisibility checks for 7 and 10. |
Divisibility of Numbers between 71 and 80Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
71 | Take out the last digit, multiply it by 7, and then subtract it from the remaining digits. | ||
72 | Check the divisibility of 8 and 9. | ||
73 | Form the alternating sum of four digits as a group from right to left. Afterwards, take out the last digit, multiply it by 22, and then add it to the remaining digits. | ||
74 | Check the divisibility of 2 and 37. | ||
75 | Check the divisibility of 3 and 25. | ||
76 | Check the divisibility of 4 and 19. | ||
77 | Check the divisibility of 7 and 11. | ||
78 | Check the divisibility of 2 and 39. | ||
79 | Take out the last digit, multiply it by 8, and then add it to the remaining digits. | ||
80 | Check the last 4 digits and determine its divisibility by 80. |
Divisibility of Numbers between 81 and 90Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
81 | Take out the last digit, multiply it by 8, and then subtract it from the remaining digits. | ||
82 | Check the divisibility of 2 and 41. | ||
83 | Take out the last digit, multiply it by 25, and then add it to the remaining digits. | ||
84 | Check the divisibility of 3, 4, and 7. | ||
85 | Check the divisibility of 5 and 17. | ||
86 | Check the divisibility of 2 and 43. | ||
87 | Check the divisibility of 3 and 29. | ||
88 | Check the divisibility of 8 and 11. | ||
89 | Take out the last digit, multiply it by 9, and then add it to the remaining digits. | ||
90 | Check the divisibility of 9 and 10. |
Divisibility of Numbers between 91 and 100Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
91 | Form the alternating sum in groups of three from right to left. Afterwards, take out the last digit, multiply it by 9, and then subtract it from the remaining digits. | ||
92 | Check the divisibility of 4 and 23. | ||
93 | Check the divisibility of 3 and 31. | ||
94 | Check the divisibility of 2 and 47. | ||
95 | Check the divisibility of 5 and 19. | ||
96 | Check the divisibility of 3 and 32 separately. | ||
97 | Take out the leftmost digit and multiply it by 3. Move that number two spaces to the right, and then add it to the remaining digits. | ||
98 | Take out the leftmost digit and multiply it by 2. Move that number two spaces to the right, and then add it to the remaining digits. | ||
99 | Add up all the digits in groups of two from right to left. | ||
100 | Examine the last 2 digits. It is divisible by 100 when the number ends in 00. |
Divisibility of Numbers between 101 and 110Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
101 | Take out the last digit, multiply it by 10, and then subtract it from the remaining digits. | ||
102 | Check the divisibility of 6 and 17. | ||
103 | Subtract the last two digits from 3 times the rest, and check if it is divisible by 103. If the result is not easy to determine, subtract 31 times the last digit from the rest of the digits. | ||
104 | Check the divisibility of 8 and 13. | ||
105 | Check the divisibility of 3, 5, and 7. | ||
106 | Check the divisibility of 2 and 53. | ||
107 | Take out the last digit, multiply it by 32, and then subtract it from the remaining digits. | ||
108 | Check the divisibility of 4 and 27. | ||
109 | Take out the last digit, multiply it by 11, and then add it to the remaining digits. | ||
110 | Check the divisibility of 10 and 11. |
Divisibility of Numbers between 111 and 120Edit
Divisor | Main Method (Easier) | Examples (for Main Method) | |
---|---|---|---|
Primary methods Alternative methods | |||
111 | Add up all the digits in groups of three from right to left. | ||
112 | Check the divisibility of 7 and 16. | ||
113 | Take out the last digit, multiply it by 34, and then add it to the remaining digits. | ||
114 | Check the divisibility of 6 and 19. | ||
115 | Check the divisibility of 5 and 23. | ||
116 | Check the divisibility of 4 and 29. | ||
117 | Check the divisibility of 9 and 13. | ||
118 | Check the divisibility of 2 and 59. | ||
119 | Check the divisibility of 7 and 17. | ||
120 | Check the divisibility of 3 and 40. |
Techinical Divisibility of Number InformationEdit
Below lists down all the possible methods. The methods recommended will be bolded and displayed in blue text.
Prime numbers from 2 ~ 100Edit
Divisor | Simple methods Advanced methods | Examples | |||
---|---|---|---|---|---|
Last digit(s) | Digit sum | Blocks of digit sum | Combined methods | ||
7 | |||||
11 | |||||
13 | |||||
17 | |||||
19 | |||||
23 | |||||
29 | |||||
31 | |||||
37 | |||||
41 | |||||
43 | |||||
47 | |||||
53 | |||||
59 | |||||
61 | |||||
67 | |||||
71 | |||||
73 | |||||
79 | |||||
83 | |||||
89 | |||||
97 | |||||
101 | |||||
103 | |||||
107 | |||||
109 | |||||
113 |