This page is an excerpt from this page. I apologize for the lack of neatness, as I do not wish to make neatness modifications.
Divisor | Divisibility condition | Examples |
---|---|---|
1 | No special condition. Any integer is divisible by 1. | 2 is divisible by 1. |
2 | The last digit is even (0, 2, 4, 6, or 8). | 1,294: 4 is even. |
3 | Sum the digits. If the result is divisible by 3, then the original number is divisible by 3. | 405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3. 16,499,205,854,376 → 1+6+4+9+9+2+0+5+8+5+4+3+7+6 sums to 69 → 6 + 9 = 15 → 1 + 5 = 6, which is clearly divisible by 3. |
Subtract the quantity of the digits 2, 5, and 8 in the number from the quantity of the digits 1, 4, and 7 in the number. | Using the example above: 16,499,205,854,376 has four of the digits 1, 4 and 7 and four of the digits 2, 5 and 8; ∴ Since 4 − 4 = 0 is a multiple of 3, the number 16,499,205,854,376 is divisible by 3. | |
4 | Examine the last two digits. | 40832: 32 is divisible by 4. |
If the tens digit is even, the ones digit must be 0, 4, or 8. If the tens digit is odd, the ones digit must be 2 or 6. |
40832: 3 is odd, and the last digit is 2. | |
Twice the tens digit, plus the ones digit. | 40832: 2 × 3 + 2 = 8, which is divisible by 4. | |
5 | The last digit is 0 or 5. | 495: the last digit is 5. |
6 | It is divisible by 2 and by 3. | 1,458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6. |
7 | Form the alternating sum of blocks of three from right to left. | 1,369,851: 851 − 369 + 1 = 483 = 7 × 69 |
Subtract 2 times the last digit from the rest. (Works because 21 is divisible by 7.) | 483: 48 − (3 × 2) = 42 = 7 × 6. | |
Or, add 5 times the last digit to the rest. (Works because 49 is divisible by 7.) | 483: 48 + (3 × 5) = 63 = 7 × 9. | |
Or, add 3 times the first digit to the next. (This works because 10a + b − 7a = 3a + b − last number has the same remainder) | 483: 4×3 + 8 = 20 remainder 6, 6×3 + 3 = 21. | |
Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, -1, -3, -2 (repeating for digits beyond the hundred-thousands place). Then sum the results. | 483595: (4 × (-2)) + (8 × (-3)) + (3 × (-1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7. | |
8 | ||
If the hundreds digit is odd, examine the number obtained by the last two digits plus 4. | 352: 52 + 4 = 56. | |
Add the last digit to twice the rest. | 56: (5 × 2) + 6 = 16. | |
Examine the last three digits. | 34152: Examine divisibility of just 152: 19 × 8 | |
Add four times the hundreds digit to twice the tens digit to the ones digit. | 34152: 4 × 1 + 5 × 2 + 2 = 16 | |
9 | Sum the digits. If the result is divisible by 9, then the original number is divisible by 9. | 2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9. |
10 | The last digit is 0. | 130: the last digit is 0. |
11 | Form the alternating sum of the digits. | 918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22. |
Add the digits in blocks of two from right to left. | 627: 6 + 27 = 33. | |
Subtract the last digit from the rest. | 627: 62 − 7 = 55. | |
Add the last digit to the hundredth place (add 10 times the last digit to the rest). | 627: 62 + 70 = 132. | |
If the number of digits is even, add the first and subtract the last digit from the rest. | 918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11 | |
If the number of digits is odd, subtract the first and last digit from the rest. | 14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 11 | |
12 | It is divisible by 3 and by 4. | 324: it is divisible by 3 and by 4. |
Subtract the last digit from twice the rest. | 324: 32 × 2 − 4 = 60. | |
13 | Form the alternating sum of blocks of three from right to left. | 2,911,272: −2 + 911 − 272 = 637 |
Add 4 times the last digit to the rest. | 637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13. | |
Subtract 9 times the last digit from the rest. | 637: 63 - 7 × 9 = 0. | |
14 | It is divisible by 2 and by 7. | 224: it is divisible by 2 and by 7. |
Add the last two digits to twice the rest. The answer must be divisible by 14. | 364: 3 × 2 + 64 = 70. 1764: 17 × 2 + 64 = 98. | |
15 | It is divisible by 3 and by 5. | 390: it is divisible by 3 and by 5. |
16 | ||
If the thousands digit is odd, examine the number formed by the last three digits plus 8. | 3,408: 408 + 8 = 416. | |
Add the last two digits to four times the rest. | 176: 1 × 4 + 76 = 80.
1168: 11 × 4 + 68 = 112. | |
Examine the last four digits. | 157,648: 7,648 = 478 × 16. | |
17 | Subtract 5 times the last digit from the rest. | 221: 22 − 1 × 5 = 17. |
18 | It is divisible by 2 and by 9. | 342: it is divisible by 2 and by 9. |
19 | Add twice the last digit to the rest. | 437: 43 + 7 × 2 = 57. |
20 | It is divisible by 10, and the tens digit is even. | 360: is divisible by 10, and 6 is even. |
If the number formed by the last two digits is divisible by 20. | 480: 80 is divisible by 20. | |
21 | Subtract twice the last digit from the rest. | 168: 16 − (8×2) = 0, 168 is divisible. 1050: 105 − (0×2) = 105, 10 − (5×2) = 0, 1050 is divisible. |
Sum of the digits are divisible by 3 and the number is divisible by 7 | 105: 1 + 0 + 5 = 6, 10 + (5×5) = 35 = 7 × 5, 105 is divisible. 210: 2 + 1 + 0 = 3, 21 + (0×5) = 21 = 7 × 3, 210 is divisible | |
23 | Add 7 times the last digit to the rest. | 3128: 312 + (8×7) = 368, 368 ÷ 23 = 16. |
25 | The number formed by the last two digits is divisible by 25. | 134,250: 50 is divisible by 25. |
27 | Sum the digits in blocks of three from right to left. If the result is divisible by 27, then the number is divisible by 27. | 2,644,272: 2 + 644 + 272 = 918 = 27×34. |
Subtract 8 times the last digit from the rest. | 621: 62 − (1×8) = 54. | |
29 | Add three times the last digit to the rest. | 261: 1×3 = 3; 3 + 26 = 29 |
31 | Subtract three times the last digit from the rest. | 837: 83 − 3×7 = 62 |
32 | ||
If the ten thousands digit is odd, examine the number formed by the last four digits plus 16. | 254,176: 4176+16 = 4192. | |
Add the last two digits to 4 times the rest. | 1,312: (13×4) + 12 = 64. | |
33 | Add 10 times the last digit to the rest; it has to be divisible by 3 and 11. | 627: 62 + 7 × 10 = 132, 13 + 2 × 10 = 33. |
Add the digits in blocks of two from right to left. | 2,145: 21 + 45 = 66. | |
35 | Number must be divisible by 7 ending in 0 or 5. | 700 is divisible by 7 ending in a 0. |
37 | Take the digits in blocks of three from right to left and add each block, just as for 27. | 2,651,272: 2 + 651 + 272 = 925. 925 = 37×25. |
Subtract 11 times the last digit from the rest. | 925: 92 − (5×11) = 37. | |
39 | Add 4 times the last digit to the rest. | 351: 1 × 4 = 4; 4 + 35 = 39 |
41 | Subtract 4 times the last digit from the rest. | 738: 73 − 8 × 4 = 41. |
43 | Add 13 times the last digit to the rest. | 36,249: 3624 + 9 × 13 = 3741, 374 + 1 × 13 = 387, |
Subtract 30 times the last digit from the rest. | 36,249: 3624 - 9 × 30 = 3354, 335 - 4 × 30 = 215 = 43 × 5. | |
45 | The number must be divisible by 9 ending in 0 or 5. | 495: 4 + 9 + 5 = 18, 1 + 8 = 9; (495 is divisible by both 5 and 9.) |
47 | Subtract 14 times the last digit from the rest. | 1,642,979: 164297 − 9 × 14 = 164171, 16417 − 14 = 16403, |
49 | Add 5 times the last digit to the rest. | 1,127: 112+(7×5)=147. 147: 14 + (7×5) = 49 |
50 | The last two digits are 00 or 50. | 134,250: 50. |
51 | Subtract 5 times the last digit to the rest. | 204: 20-(4×5)=0 |
53 | Add 16 times the last digit to the rest. | 3657: 365+(7×16)=477 = 9 × 53 |
55 | Number must be divisible by 11 ending in 0 or 5. | 935: 93 − 5 = 88 or 9 + 35 = 44. |
59 | Add 6 times the last digit to the rest. | 295: 5×6 = 30; 30 + 29 = 59 |
61 | Subtract 6 times the last digit from the rest. | 732: 73-(2×6)=61 |
64 | The number formed by the last six digits must be divisible by 64. | 2,640,000 is divisible by 64. |
65 | Number must be divisible by 13 ending in 0 or 5. | 130 is divisible by 13 ending in 0. |
66 | Number must be divisible by 6 and 11. | 132 is divisible by 6 and 11. |
67 | Subtract twice the last two digits from the rest. | 9112: 2×12 = 24; 91 - 24 = 67 |
69 | Add 7 times the last digit to the rest. | 345: 5×7 = 35; 35 + 34 = 69 |
71 | Subtract 7 times the last digit from the rest. | 852: 85-(2×7)=71 |
75 | Number must be divisible by 3 ending in 00, 25, 50 or 75. | 825: ends in 25 and is divisible by 3. |
77 | Form the alternating sum of blocks of three from right to left. | 76,923: 923 - 76 = 847. |
79 | Add 8 times the last digit to the rest. | 711: 1×8 = 8; 8 + 71 = 79 |
81 | Subtract 8 times the last digit from the rest. | 162: 16-(2×8)=0 |
83 | Add 25 times the last digit from the rest. | 581: 58+(1×25)=83 |
89 | Add 9 times the last digit to the rest. | 801: 1×9 = 9; 80 + 9 = 89 |
91 | Subtract 9 times the last digit from the rest. | 182: 18-(2×9)=0 |
Form the alternating sum of blocks of three from right to left. | 5,274,997: 5 - 274 + 997 = 728 | |
99 | Add the digits in blocks of two from right to left. | 144,837: 14 + 48 + 37 = 99. |
100 | Ends with at least two zeros. | 900 ends with 2 zeros |
101 | Form the alternating sum of blocks of two from right to left. | 40,299: 4 - 2 + 99 = 101. |
111 | Add the digits in blocks of three from right to left. | 1,370,184: 1 + 370 + 184 = 555 |
125 | The number formed by the last three digits must be divisible by 125. | 2125 is divisible by 125. |
128 | The number formed by the last seven digits must be divisible by 128. | 11,280,000 is divisible by 128. |
143 | Form the alternating sum of blocks of three from right to left. | 1,774,487: 1 - 774 + 487 = -286 |
256 | The number formed by the last eight digits must be divisible by 256. | 225,600,000 is divisible by 256. |
333 | Add the digits in blocks of three from right to left. | 410,922: 410 + 922 = 1,332 |
512 | The number formed by the last nine digits must be divisible by 512. | 1,512,000,000 is divisible by 512. |
989 | Add the last three digits to eleven times the rest. | 21758: 21 × 11 = 231; 758 + 231 = 989 |
999 | Add the digits in blocks of three from right to left. | 999,999: 999 + 999 = 1,998 |
1000 | Ends with at least three zeros. | 2000 ends with 3 zeros |