Divisibility Rules for numbers 21 to 29

Divisibility Rules for numbers 21 – 29

The divisibility rules for numbers 21 to 29 test whether any of the numbers 21 to 29 is a divisor of a given number. That is, given any number, you should be able to check or ascertain that the number is divisible by a number between 20 and 30. Below are the rules

The rule for number 21:

 (a) A number N is divisible by 21 if the difference between the remaining number and 2 times the last digit is 0 or a number divisible by 21

(b) A number N is divisible by 21 if the difference between 11 times the remaining number and the last digit is 0 or a number divisible by 21.

(c) A number N is divisible by 21 if the sum of 3 times the last digit and 9 times the remaining number is a number divisible by 21

Examples:

Example (a): Test whether 1029 is divisible by 21,                                                                                             102 – (2×9) = 102 – 18 = 84, a number divisible by 21.

Example (b): Test whether 42 is divisible by 21,                                                                                                 42:  (11 x 4) – 2 = 42 = 2 x 21

Example (c): Test whether 126 is divisible by 21,                                                                                              126: 9 x 12 + 3 x 6 = 126 = 21 x 6

The Rule for number 22

 (a) A number N is divisible by 22 if the difference between 12 times the remaining number and the last digit is 0 or a number divisible by 22.

(b) A number N is divisible by 22 if the difference between twice the remaining number and 2 times the last digit is 0 or a number divisible by 22.

(c) A number N is divisible by 21 if the sum of 3 times the last digit and 8 times the remaining number is a  number divisible by 22.

Examples for Rule 22

 Example (a) Subtracting the last digit from 12 times the rest, gives a multiple of 22,                                              110:  (12 x 11) – 0 = 132 = 22 x 6

Example (b) Subtracting twice the last digit from twice the rest, gives a multiple of 22,                                         132: 2 x 13 – 2 x 2 = 22 = 1 x  22

Example (c) Adding 3 times the last digit to 8 times the rest, gives a multiple of 22,                                          132: 8 x 13 + 3 x 2 = 110 = 22 x 5

 

The Rule for number 23

(a) A number N is divisible by 23 if the sum of 7 times the remaining number minus 3 times the last digit is a number divisible by 23.

(b) A number N is divisible by 23 if the difference between 13 times the remaining number and the last digit is 0 or a number divisible by 23

(c) A number N is divisible by 23 if the difference between of 3 times the remaining number and 2 times the last digit is a number divisible by 23.

Examples for Rule 23

Example (a) Add 3 times the last digit to 7 times the rest,                                                                                    92: 7 x 9 + 3 x 2 = 69 = 23 x 3

Example (b) Subtract the last digit from 13 times the rest,                                                                                 46:  (13 x 4) – 6 = 46 = 23 x 2

Example (c) Subtract twice the last digit from three times the rest,                                                                    138: 3 x 13 – 2 x 8 = 23 = 1 x 23

 

The rule for number 24

(a) A number N is divisible by 24 if 14 times the remaining number minus the last digit is a number divisible by 24.

(b) A number N is divisible by 24 if the difference between 4 times the remaining number and 2 times the last digit is a number divisible by 24.

(c) A number N is divisible by 24 if the sum of 6 times the remaining number and 3 times the last digit is a number divisible by 24.

Examples for Rule 24

Example (a) Test whether 48 is divisible by 24 by using  rule 24(a)                                                                    48:  (14 x 4) – 8 = 48 = 24 x 2

Example (b) Test whether 168 is divisible by 24 by using  rule 24(b)                                                                168: 4 x 16 – 2 x 8 = 48 = 2 x 24

Example (c) Test whether 144 is divisible by 24 by using  rule 24(c)                                                                 144: 6 x 14 + 3 x 4 = 96 = 24 x 4

 

The rule for number 25:

(a) A number N is divisible by 25 if the difference between 15 times the remaining number and the last digit is 0 or a number divisible by 25.

(b) A number N is divisible by 25 if the sum of 5 times the remaining number and the 3 times the last digit is a number divisible by 25.

(c) A number N is divisible by 25 if the difference between 5 times the remaining number and 2 times the last digit is 0 or a number divisible by 25.

Examples for Rule 25

Example (a)  Test whether 75 is divisible by 25 by using rule 25(a)                                                                    75:  (15 x 7) – 5 = 100 = 25 x 4

Example (b) Add 3 times the last digit to 5 times the rest,                                                                                  175: 5 x 17 + 3 x 5 = 100 = 25 x 4

Example (c) Subtract twice the last digit from 5 times the rest,                                                                         125: 5 x 12 – 2 x 5 = 50 = 2 x 25

 

The rule for number 26

(a) A number N is divisible by 26 if the sum of 6 times the remaining number minus 2 times the last digit is a number is divisible by 26.

(b) A number N is divisible by 26 if the sum of 4 times the remaining number and 3 times the last digit is a number is divisible by 26.

(c) A number N is divisible by 26 if the difference between 16 times the remaining number and the last digit is 0 or a number divisible by 26.

Examples for Rule 26

Example (a)  Test whether 104 is divisible by 26 by using rule 26(a)                                                                  104: 6x 10 – 2 x 4 = 52 = 2 x 26

Example (b)  Test whether 156 is divisible by 26 by using rule 26(b)                                                                  156: 4 x 15 + 3 x 6 = 78 = 26 x 3

Example (c)  Test whether 52 is divisible by 26 by using rule 26(c)                                                                    52:  (16 x 5) – 2 = 78 = 26 x 3

 

The rule for number 27

(a) A number N is divisible by 27 if the difference between 17 times the remaining number and the last digit is 0 or a number divisible by 27.

(b) A number N is divisible by 27 if 7 times the remaining number minus 2 times the last digit is a number divisible by 27.

(c) A number N is divisible by 27 if the sum of 3 times the last digit and 3 times the remaining number is a number divisible by 27.

Examples for Rule 27

Example (a)  Test whether 54 is divisible by 27 by using rule 27(a)                                                                   54:  (17 x 5) – 4 = 81 = 27 x 3

Example (b)  Test whether 108 is divisible by 27 by using rule 27(b)                                                                108: 7 x 10 – 2 x 8 = 54 = 2 x 27

Example (c)  Test whether 135 is divisible by 27 by using rule 27(c)                                                                 135: 3 x 13 + 3 x 5 = 54 = 27 x 2

 

The Rule for number 28

(a) A number N is divisible by 28 if the sum of the last digit and 3 times the remaining number is divisible by 28.

(b) A number N is divisible by 28 if 18 times the remaining number minus the last digit is divisible by 28.

(c) A number N is divisible by 28 if the difference between 8 times the remaining number and twice the last digit is 0 or a number divisible by 28.

Examples for Rule 28

Example (a)Test whether 168 is divisible by 28                                                                                                    68: 2 x 16 + 3 x 8 = 56 = 28 x 2

Example (a) Test whether 56 is divisible by 28                                                                                                         56:  (18 x 5) – 6 = 84 = 28 x 3

Example (a) Test whether 84 is divisible by 28                                                                                                       84: 8 x 8 – 2 x 4 = 56 = 28 x 2

 

The Rule for number 29

(a) A number N is divisible by 29 if the difference between 19 times the remaining number and the last digit is a number is divisible by 29.

(b) A number N is divisible by 29 if the difference between 9 times the remaining number and the 2 times the last digit is 0 or a number divisible by 29.

Examples for Rule 29

Example (a) Subtract the last digit from 19 times the rest, gives a multiple of 29,                                                  58:  (19 x 5) – 8 = 87 = 29 x 3

Example (b) Test whether 87 is divisible by 29 by subtracting twice the last digit from 9 times the rest,                 87: 9 x 8 – 2 x 7 = 58 = 2 x 29

 

New Rule for Divisibility of 8

Old- A number is divisible by 8 if the last 3 digits is divisible by 8.

This rule is actually difficult to test because students are required to know the multiples of 8 between 100 and 1000 that is three-digit numbers.

Proposed New Rule: A number is divisible by 8 if the SUM of the number formed by the last two digits and 4 times the hundred digit is divisible by 8

Examples: To test whether 8 can divide the following numbers- 112, 536, 708, 7152 .

  1. To test 112, separate 112 into 1 and 12. Multiply 1 by 4 and add it to 12 which now gives 4 + 12 = 16. Since 16 is divisible by 8, then 112 is divisible by 8
  1. To test 536, separate 536 into 5 and 36. Multiply 5 by 4 and add it to 36 which gives 20 + 36 = 56 a number divisible by 8. Therefore 536 is divisible by 8.
  1. To test 708, separate 708 into 7 and 08. Multiply 7 by 4 and add it to 8 which gives 28 + 8 = 36 a number not divisible by 8. Therefore 708 is not divisible by 8.
  1. To test 7152, test only the last 3 digits – that is 152. Separate 152 into 1 and 52. Multiply 1 by 4 and add it to 52 which gives 4 + 52 = 56 a number divisible by 8. Therefore 7152 is divisible by 8.

 


Divisibility Rules for numbers 2 to 10

Introduction

Divisibility rules involve finding out without actually dividing whether a given whole number is a factor of another whole number.

Many textbooks that have this topics states the following rules:

Any whole number is exactly divisible by:

   – 2 if its last digit is even
   – 3 if the sum if its digits form a number divisible by 3
   – 4 if the last two digits form a number divisible by 4
   – 5 if the last digit is 5 or 0
   – 6 If its last digit is even and the sum of its digits form a number divisible by 3
   – 8 if the last three digits form a number divisible by 8
   – 9 if the sum of the digits is divisible by 9
   – 10 if the last digit is 0