BradJolly

Easy divisibility rules that hardly anybody knows (Part 3)

Blog Post created by BradJolly Employee 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

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.


Outcomes