### 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 digi**t 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

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