on Apr 3, 2018The first and second blog posts in this series described divisibility rules for 11, 7, and 21. To summarize:
Divisor | Rule | Example |
11 | Chop, subtract | 3476 347 – 6 = 341 34 – 1 = 33 |
7 or 21 | Chop, double, subtract | 2954 295 – 8 = 287 28 – 14 = 14 |
These rules work because adding or subtracting multiples of n does not change whether a quantity is divisible by n. The “chop, subtract” procedure subtracts a multiple of 11, and the “chop, double, subtract” procedure subtracts a multiple of 21 because it subtracts from the ones digit and TWICE from the tens digit (1+2×10 = 21). Every multiple of 21 is also a multiple of 7, so “chop, double, subtract” works for both 7 and 21.
If “chop, subtract” subtracts a multiple of 11, and “chop, double, subtract” subtracts a multiple of 21, then “chop, triple, subtract” and “chop, quadruple, subtract” subtract multiples of 31 and 41. Therefore, we can expand our list of divisibility rules as shown below.
Divisor | Rule | Example |
11 | Chop, subtract | 3476 347 – 6 = 341 34 – 1 = 33 |
7 or 21 | Chop, double, subtract | 2954 295 – 8 = 287 28 – 14 = 14 |
31 | Chop, ×3, subtract | 1023 102 – 9 = 93 9 – 9 = 0 |
41 | Chop, ×4, subtract | 2132 213 – 8 = 205 20 – 20 = 0 |
17 or 51 | Chop, ×5, subtract | 6987 698 – 35 = 663 66 – 15 = 51 5 – 5 = 0 |
61 | Chop, ×6, subtract | 3172 317 – 12 = 305 30 – 30 = 0 |
71 | Chop, ×7, subtract | 6532 653 – 14 = 639 63 – 63 = 0 |
27 or 81 | Chop, ×8, subtract | 2214 221 – 32 = 189 18 – 72 = 54* |
13 or 91 | Chop, ×9, subtract | 2119 211 – 81 = 130 |
101 | Chop, ×10, subtract | 15,352 1535 – 20 = 1515 151 – 50 = 101 |
37 or 111 | Chop, ×11, subtract | 7992 799 – 22 = 777 77 – 77 = 0 |
121 | Chop, ×12, subtract | 9801 980 – 12 = 968 96 – 96 = 0 |
67 or 201 | Chop, ×20, subtract | 5427 542 – 140 = 402 40 – 40 = 0 |
43 or 301 | Chop, ×30, subtract | 1376 137 – 180 = 43* |
401 | Chop, ×40, subtract | 11,228 1122 – 320 = 802 |
167 or 501 | Chop, ×50, subtract | 10,354 1035 – 200 = 835 83 – 250 = 167* |
* In these algorithms, subtraction always yields the positive difference.
Note that some rules apply to multiple numbers. That is because the rule subtracts a multiple of the larger number, and the smaller number is a factor of the larger number. For example, the “Chop, ×5, subtract” rule applies to 17 and 51 because the rule subtracts multiples of 51 and every multiple of 51 is also a multiple of 17.
The rules above go well beyond the "2, 3, 4, 5, 6, 9, 10" list that students typically learn. They are easy to remember and to understand, and they deepen students' understanding of why divisibility algorithms work in the first place. Parents and teachers should consider exposing students to these rules.