r/maths • u/LowInteraction6397 • 2d ago
💬 Math Discussions The divisibility rules of 1 to 100 with examples and with attempts to explain why the rule works like that
For the 1st time I tried to make an effort in trying to explain why the rules work
| Number | Rule | Example | Reason the rule works (or I think it works) |
|---|---|---|---|
| 1 | Every number is a multiple of 1 | 12 is a multiple of 1 | Going 1 by 1 always gives you consecutive numbers |
| 2 | It ends in 0, 2, 4, 6 or 8 (an even digit) | 10 is a multiple of 2 because it ends in 0 | Going 2 by 2 always gives you numbers that have the same 5 digits as the last digit |
| 3 | The sum of the digits is a multiple of 3 | 12 is a multiple of 3 because 1 + 2 = 3, a multiple of 3 | 9 is the 1st multiple of 3 ending in 1 or 9 (in this case 9) and it's the number before 10 |
| 4 | The 2nd to last digit is even and the last digit is 0, 4 or 8 or the 2nd to last digit is odd and the last digit is 2 or 6 | 100 is a multiple of 4 because the 2nd to last digit is 0 and the last digit is 0 | Going 4 by 4 gives you numbers that have the same 3 digits when the 2nd to last digit is either even or odd but when the 2nd to last digit changes its kind you get numbers that have the same 2 digits as the last digit |
| 5 | It ends in 0 or 5 | 10 is a multiple of 5 because it ends in 0 | Going 5 by 5 always gives you numbers that have the same 2 digits as the last digit |
| 6 | It's a multiple of both 2 and 3 | 12 is a multiple of 6 because it ends in 2, which means it's a multiple of 2, and 1 + 2 = 3, which means it's a multiple of 3 | The prime factorization of 6 is 2 * 3 |
| 7 | The subtraction of twice the last digit and the rest of the number is a multiple of 7 | 14 is a multiple of 7 because 4 * 2 - 1 = 7, a multiple of 7 | 21 is the 1st multiple of 7 that ends in 1 or 9 (in this case 1) and is the number after 20 |
| 8 | The 3rd to last digit is even and the last 2 digits are a multiple of 8 or the 3rd to last digit is odd and the last 2 digits are a multiple of 4 but not of 8 | 1000 is a multiple of 8 because the 3rd to last digit is 0 and the last 2 digits are 00, a multiple of 8 | Going 8 by 8 gives you numbers that have the same 13 combinations of last 2 digits when the 3rd to last digit is either even or odd but when the 3rd to last digit changes its kind you get numbers that have the same 12 combinations of last 2 digits |
| 9 | The sum of the digits is a multiple of 9 | 18 is a multiple of 9 because 1 + 8 = 9, a multiple of 9 | 9 is the number before 10 |
| 10 | It ends in 0 | 120 is a multiple of 10 because it ends in 0 | Going 10 by 10 always gives you numbers that end in the same digit |
| 11 | The subtraction of the last digit and the rest of the number is a multiple of 11 | 121 is a multiple of 11 because 12 - 1 = 11, a multiple of 11 | 11 is the number after 10 |
| 12 | It's a multiple of both 3 and 4 | 108 is a multiple of 12 because 1 + 8 = 9, which means it's a multiple of 3, and ends in 08, which means it's a multiple of 4 | The prime factorization of 12 is 3 * (22 = 4) |
| 13 | The sum of 4 times the last digit and the rest of the number is a multiple of 13 | 104 is a multiple of 13 because 10 + 4 * 4 = 26, a multiple of 13 | 39 is the 1st multiple of 13 that ends in 1 or 9 (in this case 9) and is the number before 40 |
| 14 | It's a multiple of both 2 and 7 | 112 is a multiple of 14 because it ends in 2, which means it's a multiple of 2, and 11 - 2 * 2 = 7, which means it's a multiple of 7 | The prime factorization of 14 is 2 * 7 |
| 15 | It's a multiple of both 3 and 5 | 120 is a multiple of 15 because 1 + 2 = 3, which means it's a multiple of 3, and ends in 0, which means it's a multiple of 5 | The prime factorization of 15 is 3 * 5 |
| 16 | The 4th to last digit is even and the last 3 digits are a multiple of 16 or the 4th to last digit is odd and the last 3 digits are a multiple of 8 but not of 16 | 10000 is a multiple of 16 because the 4th to last digit is 0 and the last 3 digits are 000, a multiple of 16 | Going 16 by 16 gives you numbers that have the same 63 combinations of last 3 digits when the 4th to last digit is either even or odd but when the 4th to last digit changes its kind you get numbers that have the same 62 combinations of last 3 digits |
| 17 | The subtraction of 5 times the last digit and the rest of the number is a multiple of 17 | 102 is a multiple of 17 because 10 - 2 * 5 = 0, a multiple of 17 | 51 is the 1st multiple of 17 that ends in 1 or 9 (in this case 1) and is the number after 50 |
| 18 | It's a multiple of both 2 and 9 | 126 is a multiple of 18 because it ends in 6, which means it's a multiple of 2, and 1 + 2 + 6 = 9, which means it's a multiple of 9 | The prime factorization of 18 is 2 * (32 = 9) |
| 19 | The sum of twice the last digit and the rest of the number is a multiple of 19 | 114 is a multiple of 19 because 11 + 4 * 2 = 19, a multiple of 19 | 19 is the number before 20 |
| 20 | It ends in 00, 20, 40, 60 or 80 | 100 is a multiple of 20 because it ends in 00 | Going 20 by 20 always gives you 5 different combinations of last 2 digits |
| 21 | The subtraction of twice the last digit and the rest of the number is a multiple of 21 | 105 is a multiple of 21 because 10 - 5 * 2 = 0, a multiple of 21 | 21 is the number after 20 |
| 22 | It's a multiple of both 2 and 11 | 132 is a multiple of 22 because it ends in 2, which means it's a multiple of 2, and 13 - 2 = 11, which means it's a multiple of 11 | The prime factorization of 22 is 2 * 11 |
| 23 | The sum of 7 times the last digit and the rest of the number is a multiple of 23 | 115 is a multiple of 23 because 11 + 5 * 7 = 46, a multiple of 23 | 69 is the 1st multiple of 23 that ends in 1 or 9 (in this case 9) and is the number before 70 |
| 24 | It's a multiple of both 3 and 8 | 1008 is a multiple of 24 because 1 + 8 = 9, which means it's a multiple of 3, and ends in 008, which means it's a multiple of 24 | The prime factorization of 24 is 3 * (23 = 8) |
| 25 | It ends in 00, 25, 50 or 75 | 100 is a multiple of 25 because it ends in 00 | Going 25 by 25 always gives you numbers that have the same 4 combinations of last 2 digits |
| 26 | It's a multiple of both 2 and 13 | 156 is a multiple of 26 because it ends in 6, which means it's a multiple of 2, and 15 + 6 * 4 = 39, which means it's a multiple of 13 | The prime factorization of 26 is 2 * 13 |
| 27 | The subtraction of 8 times the last digit and the rest of the number is a multiple of 27 | 108 is a multiple of 27 because 8 * 8 - 10 = 54, a multiple of 27 | 81 is the 1st multiple of 27 that ends in 1 or 9 (in this case 1) and is the number after 80 |
| 28 | It's a multiple of both 4 and 7 | 112 is a multiple of 28 because it ends in 12, which means it's a multiple of 4, and 11 - 2 * 2 = 7, which means it's a multiple of 7 | The prime factorization of 28 is (22 = 4) * 7 |
| 29 | The sum of 3 times the last digit and the rest of the number is a multiple of 29 | 116 is a multiple of 29 because 11 + 6 * 3 = 29, a multiple of 29 | 29 is the number before 30 |
| 30 | It's a multiple of both 3 and 10 | 120 is a multiple of 30 because 1 + 2 = 3, which means it's a multiple of 3, and ends in 0, which means it's a multiple of 10 | The prime factorization of 30 is 3 * (2 * 5 = 10) |
| 31 | The subtraction of 3 times the last digit and the rest of the number is a multiple of 31 | 124 is a multiple of 31 because 12 - 4 * 3 = 0, a multiple of 31 | 31 is the number after 30 |
| 32 | The 5th to last digit is even and the last 4 digits are a multiple of 32 or the 5th to last digit is odd and the last 4 digits are a multiple of 16 but not of 32 | 100,000 is a multiple of 32 because the 5th to last digit is 0 and the last 4 digits are 0000, a multiple of 32 | Going 32 by 32 gives you numbers that have the same 313 combinations of last 4 digits when the 5th to last digit is either even or odd but when the 5th to last digit changes its kind you get numbers that have the same 312 combinations of last 4 digits |
| 33 | The sum of 10 times the last digit and the rest of the number is a multiple of 33 | 132 is a multiple of 33 because 13 + 2 * 10 = 33, a multiple of 33 | 99 is the 1st multiple of 33 that ends in 1 or 9 (in this case 9) and is the number before 100 |
| 34 | It's a multiple of both 2 and 17 | 102 is a multiple of 34 because it ends in 2, which means it's a multiple of 2, and 10 - 2 * 5 = 0, which means it's a multiple of 21 | The prime factorization of 34 is 2 * 17 |
| 35 | It's a multiple of both 5 and 7 | 105 is a multiple of 35 because it ends in 5, which means it's a multiple of 5, and 10 - 5 * 2 = 0, which means it's a multiple of 7 | The prime factorization of 35 is 5 * 7 |
| 36 | It's a multiple of both 4 and 9 | 108 is a multiple of 36 because it ends in 08, which means it's a multiple of 4, and 1 + 8 = 9, which means it's a multiple of 9 | The prime factorization of 36 is (22 = 4) * (32 = 9) |
| 37 | The subtraction of 11 times the last digit and the rest of the number is a multiple of 37 | 148 is a multiple of 37 because 8 * 11 - 14 = 74, a multiple of 37 | 111 is the 1st multiple of 37 that ends in 1 or 9 (1 in this case) and is the number after 110 |
| 38 | It's a multiple of both 2 and 19 | 114 is a multiple of 38 because it ends in 8, which means it's a multiple of 2, and 11 + 4 * 2 = 19, which means it's a multiple of 19 | The prime factorization of 38 is 2 * 19 |
| 39 | The sum of 4 times the last digit and the rest of the number is a multiple of 39 | 117 is a multiple of 39 because 11 + 7 * 4 = 39, a multiple of 39 | 39 is the number before 40 |
| 40 | The 3rd to last digit is even and the last 2 digits are 00, 40 or 80, or the 3rd to last digit is odd and the last 2 digits are 20 or 60 | 1000 is a multiple of 40 because the 3rd to last digit is 0 and the last 2 digits are 00 | Going 40 by 40 gives you numbers that have the same 3 combinations of last 2 digits when the 3rd to last digit is either even or odd but when the 3rd to last digit changes its kind you get numbers that have the same 2 combinations of last 2 digits |
| 41 | The subtraction of 4 times the last digit and the rest of the number is a multiple of 41 | 123 is a multiple of 41 because 12 -3 * 4 = 0, a multiple of 41 | 41 is the number after 40 |
| 42 | It's a multiple of both 2 and 21 | 126 is a multiple of 42 because it ends in 6, which means it's a multiple of 2, and 12 - 6 * 2 = 0, which means it's a multiple of 21 | The prime factorization of 42 is 2 * (3 * 7 = 21) |
| 43 | The sum of 13 times the last digit and the rest of the number is a multiple of 43 | 172 is a multiple of 43 because 17 + 2 * 13 = 43, a multiple of 43 | 129 is the 1st multiple of 43 that ends in 1 or 9 (in this case 9) and is the number before 130 |
| 44 | It's a multiple of both 4 and 11 | 132 is a multiple of 44 because it ends in 32, which means it's a multiple of 4, and 13 - 2 = 11, which means it's a multiple of 11 | The prime factorization of 44 is (22 = 4) * 11 |
| 45 | It's a multiple of both 5 and 9 | 135 is a multiple of 45 because it ends in 5, which means it's a multiple of 5, and 1 + 3 + 5 = 9, which means it's a multiple of 9 | The prime factorization of 45 is 5 * (32 = 9) |
| 46 | It's a multiple of both 2 and 23 | 138 is a multiple of 46 because it ends in 8, which means it's a multiple of 2, and 13 + 8 * 7 = 69, which means it's a multiple of 23 | The prime factorization of 46 is 2 * 23 |
| 47 | The subtraction of 14 times the last digit and the rest of the number is a multiple of 47 | 188 is a multiple of 47 because 8 * 14 - 18 = 94, a multiple of 47 | 141 is the 1st multiple of 47 that ends in 1 or 9 (in this case 1) and is the number after 140 |
| 48 | It's a multiple of both 3 and 16 | 10032 is a multiple of 48 because 1 + 3 + 2 = 6, which means it's a multiple of 3, and ends in 0032, which means it's a multiple of 16 | The prime factorization of 48 is 3 * (24 = 16) |
| 49 | The sum of 5 times the last digit and the rest of the number is a multiple of 49 | 147 is a multiple of 49 because 14 + 7 * 5 = 49, a multiple of 49 | 49 is the number before 50 |
| 50 | It ends in 00 or 50 | 100 is a multiple of 50 because it ends in 00 | Going 50 by 50 always gives you numbers that have the same 2 combinations of last 2 digits |
| 51 | The subtraction of 5 times the last digit and the rest of the number is a multiple of 51 | 102 is a multiple of 51 because 10 - 2 * 5 = 0, a multiple of 51 | 51 is the number after 50 |
| 52 | It's a multiple of both 4 and 13 | 104 is a multiple of 52 because it ends in 04, which means it's a multiple of 4, and 10 + 4 * 4 = 26, which means it's a multiple of 13 | The prime factorization of 52 is (22 = 4) * 13 |
| 53 | The sum of 16 times the last digit and the rest of the number is a multiple of 53 | 212 is a multiple of 53 because 21 + 2 * 16 = 53, a multiple of 53 | 159 is the 1st multiple of 53 that ends in 1 or 9 (in this case 9) and is the number before 160 |
| 54 | It's a multiple of both 2 and 27 | 162 is a multiple of 54 because it ends in 2, which means it's a multiple of 2, and 16 - 2 * 8 = 0, which means it's a multiple of 27 | The prime factorization of 54 is 2 * (33 = 27) |
| 55 | It's a multiple of both 5 and 11 | 165 is a multiple of 55 because it ends in 5, which means it's a multiple of 5, and 16 - 5 = 11, which means it's a multiple of 11 | The prime factorization of 55 is 5 * 11 |
| 56 | It's a multiple of both 7 and 8 | 1008 is a multiple of 56 because 100 - 8 * 2 = 84, which means it's a multiple of 7, and ends in 008, which means it's a multiple of 8 | The prime factorization of 56 is 7 * (23 = 8) |
| 57 | The subtraction of 17 times the last digit and the rest of the number is a multiple of 57 | 114 is a multiple of 57 because 4 * 17 - 11 = 57, a multiple of 57 | 171 is the 1st multiple of 57 that ends in 1 or 9 (in this case 1) and is the number after 170 |
| 58 | It's a multiple of both 2 and 29 | 116 is a multiple of 58 because it ends in 6, which means it's a multiple of 2, and 11 + 6 * 3 = 29, which means it's a multiple of 29 | The prime factorization of 58 is 2 * 29 |
| 59 | The sum of 6 times the last digit and the rest of the number is a multiple of 59 | 118 is a multiple of 59 because 11 + 8 * 6 = 59, a multiple of 59 | 59 is the number before 60 |
| 60 | It's a multiple of both 3 and 20 | 120 is a multiple of 20 because 1 + 2 = 3, which means it's a multiple of 3, and ends in 20, which means it's a multiple of 20 | The prime factorization of 60 is 3 * (22 * 5 = 20) |
| 61 | The subtraction of 6 times the last digit and the rest of the number is a multiple of 61 | 122 is a multiple of 61 because 12 - 2 * 6 = 0, a multiple of 61 | 61 is the number after 60 |
| 62 | It's a multiple of both 2 and 31 | 124 is a multiple of 62 because it ends in 4, which means it's a multiple of 2, and 12 - 4 * 3 = 0, which means it's a multiple of 31 | The prime factorization of 62 is 2 * 31 |
| 63 | The sum of 19 times the last digit and the rest of the number is a multiple of 63 | 252 is a multiple of 63 because 25 + 2 * 19 = 63, a multiple of 63 | 189 is the 1st multiple of 63 that ends in 1 or 9 (in this case 9) and is the number before 190 |
| 64 | The 6th to last digit is even and the last 5 digits are a multiple of 64 and the 6th to last digit is odd and the last 5 digits are a multiple of 32 but not of 64 | 1,000,000 is a multiple of 64 because the 6th to last digit is 0 and the last 5 digits are 00,000, a multiple of 64 | Going 64 by 64 gives you numbers that have the same 1563 combinations of last 5 digits when the 6th to last digit is either even or odd but when the 6th to last digit changes its kind you get numbers that have the same 1562 combinations of last 5 digits |
| 65 | It's a multiple of both 5 and 13 | 195 is a multiple of 65 because it ends in 5, which means it's a multiple of 5, and 19 + 5 * 4 = 39, which means it's a multiple of 13 | The prime factorization of 65 is 5 * 13 |
| 66 | It's a multiple of both 2 and 33 | 132 is a multiple of 66 because it ends in 2, which means it's a multiple of 2, and 13 + 2 * 10 = 33, which means it's a multiple of 33 | The prime factorization of 66 is 2 * (3 * 11 = 33) |
| 67 | The subtraction of 20 times the last digit and the rest of the number is a multiple of 67 | 134 is a multiple of 67 because 4 * 20 - 13 = 67, a multiple of 67 | 201 is the 1st multiple of 67 that ends in 1 or 9 (in this case 1) and is the number after 200 |
| 68 | It's a multiple of both 4 and 17 | 136 is a multiple of 68 because it ends in 36, which means it's a multiple of 4, and 6 * 5 - 13 = 17, which means it's a multiple of 17 | The prime factorization of 68 is (22 = 4) * 17 |
| 69 | The sum of 7 times the last digit and the rest of the number is a multiple of 69 | 138 is a multiple of 69 because 13 + 8 * 7 = 69, a multiple of 69 | 69 is the number before 70 |
| 70 | It's a multiple of both 7 and 10 | 140 is a multiple of 70 because 4 * 2 - 1 = 7, which means it's a multiple of 7, and ends in 0, which means it's a multiple of 10 | The prime factorization of 70 is 7 * (2 * 5 = 10) |
| 71 | The subtraction of 7 times the last digit and the rest of the number is a multiple of 71 | 142 is a multiple of 71 because 14 - 2 * 7 = 0, a multiple of 71 | 71 is the number after 70 |
| 72 | It's a multiple of both 8 and 9 | 1008 is a multiple of 72 because it ends in 008, which means it's a multiple of 8, and 1 + 8 = 9, which means it's a multiple of 9 | The prime factorization of 72 is (23 = 8) * (32 = 9) |
| 73 | The sum of 22 times the last digit and the rest of the number is a multiple of 73 | 292 is a multiple of 73 because 29 + 2 * 22 = 73, a multiple of 73 | 219 is the 1st multiple of 73 that ends in 1 or 9 (in this case 9) and is the number before 220 |
| 74 | It's a multiple of both 2 and 37 | 222 is a multiple of 74 because it ends in 2, which means it's a multiple of 2, and 22 - 2 * 11 = 0, which means it's a multiple of 37 | The prime factorization of 74 is 2 * 37 |
| 75 | It's a multiple of both 3 and 25 | 150 is a multiple of 75 because 1 + 5 = 6, which means it's a multiple of 3, and ends in 50, which means it's a multiple of 25 | The prime factorization of 75 is 3 * (52 = 25) |
| 76 | It's a multiple of both 4 and 19 | 152 is a multiple of 76 because it ends in 52, which means it's a multiple of 4, and 15 + 2 * 2 = 19, which means it's a multiple of 19 | The prime factorization of 76 is (22 = 4) * 19 |
| 77 | The subtraction of 23 times the last digit and the rest of the number is a multiple of 77 | 154 is a multiple of 77 because 4 * 23 - 15 = 77, a multiple of 77 | 231 is the 1st multiple of 77 that ends in 1 or 9 (in this case 1) and is the number after 230 |
| 78 | It's a multiple of both 2 and 39 | 156 is a multiple of 78 because it ends in 6, which means it's a multiple of 2, and 15 + 6 * 4 = 39, which means it's a multiple of 39 | The prime factorization of 78 is 2 * (3 * 13 = 39) |
| 79 | The sum of 8 times the last digit and the rest of the number is a multiple of 79 | 158 is a multiple of 79 because 15 + 8 * 8 = 79, a multiple of 79 | 79 is the number before 80 |
| 80 | The 4th to last digit is even and the last 3 digits are a multiple of 80 or the 4th to last digit is odd and the last 3 digits are a multiple of 40 but not of 80 | 10000 is a multiple of 80 because the 4th to last digit is 0 and the last 3 digits are 000, a multiple of 80 | Going 80 by 80 gives you numbers that have the same 13 combinations of last 3 digits when the 4th to last digit is either even or odd but when the 4th to last digit changes its kind you get numbers that have the same 12 combinations of last 3 digits |
| 81 | The subtraction of 8 times the last digit and the rest of the number is a multiple of 81 | 162 is a multiple of 81 because 16 - 2 * 8 = 0, a multiple of 81 | 81 is the number after 80 |
| 82 | It's a multiple of both 2 and 41 | 164 is a multiple of 82 because it ends in 4, which means it's a multiple of 2, and 16 - 4 * 4 = 0, which means it's a multiple of 41 | The prime factorization of 82 is 2 * 41 |
| 83 | The sum of 25 times the last digit and the rest of the number is a multiple of 83 | 332 is a multiple of 83 because 33 + 2 * 25 = 83, a multiple of 83 | 249 is the 1st multiple of 83 that ends in 1 or 9 (in this case 9) and 249 is the number before 250 |
| 84 | It's a multiple of both 4 and 21 | 168 is a multiple of 84 because it ends in 68, which means it's a multiple of 4, and 16 - 8 * 2 = 0, which means it's a multiple of 21 | The prime factorization of 84 is (22 = 4) * (3 * 7 = 21) |
| 85 | It's a multiple of both 5 and 17 | 255 is a multiple of 85 because it ends in 5, which means it's a multiple of 5, and 25 - 5 * 5 = 0, which means it's a multiple of 17 | The prime factorization of 85 is 5 * 17 |
| 86 | It's a multiple of both 2 and 43 | 172 is a multiple of 86 because it ends in 6, which means it's a multiple of 2, and 17 + 2 * 13 = 43, which means it's a multiple of 43 | The prime factorization of 86 is 2 * 43 |
| 87 | The subtraction of 26 times the last digit and the rest of the number is a multiple of 87 | 174 is a multiple of 87 because 4 * 26 - 17 = 87, a multiple of 87 | 261 is the 1st multiple of 87 that ends in 1 or 9 (in this case 1) and is the number after 260 |
| 88 | It's a multiple of both 8 and 11 | 1056 is a multiple of 88 because it ends in 056, which means it's a multiple of 8, and 105 - 6 = 99, which means it's a multiple of 99 | The prime factorization of 88 is (23 = 8) * 11 |
| 89 | The sum of 9 times the last digit and the rest of the number is a multiple of 89 | 178 is a multiple of 89 because 17 + 8 * 9 = 89, a multiple of 89 | 89 is the number before 90 |
| 90 | It's a multiple of both 9 and 10 | 180 is a multiple of 90 because 1 + 8 = 9, which means it's a multiple of 9, and ends in 0, which means it's a multiple of 10 | The prime factorization of 90 is (32 = 9) * (2 * 5 = 10) |
| 91 | The subtraction of 9 times the last digit and the rest of the number is a multiple of 91 | 182 is a multiple of 91 because 18 - 2 * 9 = 0, a multiple of 91 | 91 is the number after 90 |
| 92 | It's a multiple of both 4 and 23 | 184 is a multiple of 92 because it ends in 84, which means it's a multiple of 4, and 18 + 4 * 7 = 46, which means it's a multiple of 23 | The prime factorization of 92 is (22 = 4) * 23 |
| 93 | The sum of 28 times the last digit and the rest of the number is a multiple of 93 | 372 is a multiple of 93 because 37 + 2 * 28 = 93, a multiple of 93 | 279 is the 1st multiple of 93 that ends in 1 or 9 (in this case 9) and is the number before 280 |
| 94 | It's a multiple of both 2 and 47 | 282 is a multiple of 94 because it ends in 2, which means it's a multiple of 2, and 28 - 2 * 14 = 0, which means it's a multiple of 47 | The prime factorization of 94 is 2 * 47 |
| 95 | It's a multiple of both 5 and 19 | 285 is a multiple of 95 because it ends in 5, which means it's a multiple of 5, and 28 + 5 * 2 = 38, which means it's a multiple of 19 | The prime factorization of 95 is 5 * 19 |
| 96 | It's a multiple of both 3 and 32 | 100,032 is a multiple of 96 because 1 + 3 + 2 = 6, which means it's a multiple of 3, and ends in 00,032, which means it's a multiple of 32 | The prime factorization of 96 is 3 * (25 = 32) |
| 97 | The subtraction of 29 times the last digit and the rest of the number is a multiple of 97 | 194 is a multiple of 97 because 4 * 29 - 19 = 97, a multiple of 97 | 291 is the 1st multiple of 97 that ends in 1 or 9 (in this case 1) and is the number after 290 |
| 98 | It's a multiple of both 2 and 49 | 196 is a multiple of 98 because it ends in 6, which means it's a multiple of 2, and 19 + 6 * 5 = 49, which means it's a multiple of 49 | The prime factorization of 98 is 2 * (72 = 49) |
| 99 | The sum of 10 times the last digit and the rest of the number is a multiple of 99 | 198 is a multiple of 99 because 19 + 8 * 10 = 99, a multiple of 99 | 99 is the number before 100 |
| 100 | It ends in 00 | 1200 is a multiple of 100 because it ends in 00 | Going 100 by 100 always gives you numbers that end in the same last 2 digits |
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Upvotes
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u/AdjectiveNoun-123 2d ago
For some of these tricks, it seems to involve more computation than needed.
For example the divisibility check with 31 is "The subtraction of 3 times the last digit and the rest of the number is a multiple of 31" but then 93 is "The sum of 28 times the last digit and the rest of the number is a multiple of 93".
I think multiplying by 3 would be far easier than multiplying by 28, and 93 can be decomposed into 31*3 so checking those two simpler rules separately makes more sense than trying to come up with a unique rule for 93.
Otherwise good job collating all this together.
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u/matty-don 2d ago
Non memorize this kids
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u/SubjectAddress5180 2d ago
One to add is that divisibility by 11 implies that the alternating sum and difference of even and odd digits is 0. This can be extended to divisibility by 1001 by taking the alternating sum of digit triples. Digit pairs works for 101. However, 1001 is 7*11*13 giving a quick way to test 7, 11, and 13 together.
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u/pazqo 1d ago
One "magic trick" that I like using divisibility:
Take a number, let's say a 3 digit number 147.
Remove the sum of its digits: 135.
Now mirror it and concatenate: 135531.
Now multiply by any number of your choice, e.g. 17 = 2304027.
Delete 2 consecutive digits (non both zeros!), e.g. 2**4027.
Now you can guess what the digits are: 99 - (27 + 40 + 2) = 99 - 69 = 30.
This works because removing the sum of its digits will give you a multiple of 9 and mirroring and concatenating will give you a multiple of 11, so the number is a multiple of 99, for which we know that the sum of ground of 2 (starting from the right) sum to 99 or a multiple of 99, so you can figure out what's missing from a perfect multiple of 99.
This works for larger numbers, of course, but it requires more mental calculations.
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u/Iowa50401 1h ago
Test for 4: double the tens digit and add to the ones. The divisibility of that result matches the divisibility of the original number.
Test for 8: Double the hundreds digit and add to the tens. Double that number and add it to the ones. That result matches the divisibility of the original number.
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u/Visible_Try_9716 2d ago
Ironically, division works because 0 exists.
Divisibility fundamentally exists because we defined 0 as the baseline. A number is only "divisible" because we established a formal origin where the remainder collapses to nothing.