# Easy divisibility rules that hardly anybody knows (Part 3)

Blog Post created by BradJolly on Apr 3, 2018

The first and second blog posts in this series described divisibility rules for 11, 7, and 21. To summarize:

 Divisor Rule Example 11 Chop, subtract 3476347 – 6 = 34134 – 1 = 33 7 or 21 Chop, double, subtract 2954295 – 8 = 28728 – 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 3476347 – 6 = 34134 – 1 = 33 7 or 21 Chop, double, subtract 2954295 – 8 = 28728 – 14 = 14 31 Chop, ×3, subtract 1023102 – 9 = 939 – 9 = 0 41 Chop, ×4, subtract 2132213 – 8 = 20520 – 20 = 0 17 or 51 Chop, ×5, subtract 6987698 – 35 = 66366 – 15 = 515 – 5 = 0 61 Chop, ×6, subtract 3172317 – 12 = 30530 – 30 = 0 71 Chop, ×7, subtract 6532653 – 14 = 63963 – 63 = 0 27 or 81 Chop, ×8, subtract 2214221 – 32 = 18918 – 72 = 54* 13 or 91 Chop, ×9, subtract 2119211 – 81 = 130 101 Chop, ×10, subtract 15,3521535 – 20 = 1515151 – 50 = 101 37 or 111 Chop, ×11, subtract 7992799 – 22 = 77777 – 77 = 0 121 Chop, ×12, subtract 9801980 – 12 = 96896 – 96 = 0 67 or 201 Chop, ×20, subtract 5427542 – 140 = 40240 – 40 = 0 43 or 301 Chop, ×30, subtract 1376137 – 180 = 43* 401 Chop, ×40, subtract 11,2281122 – 320 = 802 167 or 501 Chop, ×50, subtract 10,3541035 – 200 = 83583 – 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.