Home > Pre-Algebra > Factors and Multiples > Divisibility Rules

## Divisibility Rules

#### Introduction

Numbers that are divisible by one another can be divided to leave a whole number. For example, 33 is divisible by 3, because $rac{33}{3} = 11$.

34 is not divisible by 3 because $rac{34}{3} = 11.overline{3}$

Divisibility rules are useful shortcuts to work out quickly if one number can be divided by another.

#### Terms

Divisible by - When one number can be divided by another to make a whole number.
Divisibility rules - Shortcuts that let you know quickly whether a number is divisible by another.

## Lesson

Each number has its own test to see if a number is divisible by it. For example, 1 has the simplest divisibility rule of all: is the number an integer. If so, it's divisible by 1.

The divisibility rules for the numbers up to 10 are as follows:

### 2

If the last digit is even, then the number is divisible by 2.

2398798 is divisible by 2 because 8 is even.

6723643 is NOT divisible by 2, because 3 is odd.

### 3

If you add up all the digits, and the total is a multiple of three (3,6, or 9), then the number is divisible by 3.

$153 = 1+5+3 = 9$. Therefore, 153 is divisible by 3.

$151 = 1+5+1 = 7$. Therefore, 151 is NOT divisible by 3.

If you add up all the digits and get a two digit number, just add those digits together, and repeat until you get a single-digit number.

$297 = 2+9+7 = 18 = 1+8 = 9$. Therefore 297 is divisible by 3.

$295 = 2+9+5 = 16 = 1+6 = 7$. Therefore 295 is NOT divisble by 3.

### 4

If the last two digits are divisible by 4, then the whole number is too.

1716 is divisible by 4, because 16 is a multiple of four.

1718 is NOT divisible by 4, because 18 is not a multiple of 4.

### 5

If the last digit is 0 or 5 then it is divisible by 5.

8720535 is a multiple of 5.

8720534 is NOT a multiple of 5.

### 6

To work out if a number is divisible by 6 then check whether the number is even, and then apply the rule for numbers divisible by 3.

$150 = 1+5+0 = 6$. Therefore, 150 is a multiple of 6, because it is even and a multiple of 3.

$153 = 1+5+3 = 9$. Therefore, 153 is NOT* a multiple of 6, because even though it is a multiple of 3, it's not an even number.

$151 = 1+5+1 = 7$. Therefore, 151 is NOT divisible by 6.

### 7

To work out if a number is divisible by 7, you need to double the last digit and subtract it from the other digits. If this number is a multiple of 7, then the entire number is divisible by 7.

861

Double the last digit: (1 x 2 = 2)

Subtract that from the other digits (86-2 = 84)

84 is a multiple of 7, so 861 is divisible by 7

864

Double the last digit: (4 x 2 = 8)

Subtract that from the other digits (86-8 = 78)

78 is not a multiple of 7, so 864 is NOT divisible by 7

### 8

If the last three numbers are divisible by 8 then the entire number is divisible by 8. A quick check to see if a number is divisible 8 is to half it, then half it again. If it's still a whole number, then the original number is a multiple of 8.

10416. 416 is a multiple of 8, and therefore 10416 is divisible by 8.

(Half of 416 is 208, and half of that is 104. Since this is a whole number, 416 is a whole number.)

10414. 414 is NOT a multiple of 8, and therefore 10416 is not divisible by 8.

(Half of 414 is 207, and half of that is 103.5, which is not a whole number.)

### 9

To find if a number is divisible by 9 add up all the digits until you get a single digit number. If the single digit is 9, then the number is divisible by 9. If not, then the number isn't a multiple of 9.

$8172 = 8+1+7+2 = 18 = 1+8 = 9$. Therefore, 8172 is a multiple of 9.

$8176 = 8+1+7+6 = 22 = 2+2 = 4$. Therefore, 8176 is NOT a multiple of 9.

### 10

If the number ends in 0 it's divisible by 10.

## Examples

Which of the following numbers is not divisible by 2?
If a whole number ends in 0, 2, 4, 6, or 8, then the number is called an even number and is divisible by 2.
$2,233$ ($2,233$ is not even) $\rightarrow$ No
$6,772$ ($6,772$ is even) $\rightarrow$ Yes
$7,176$ ($7,176$ is even) $\rightarrow$ Yes
$9,014$ ($9,014$ is even) $\rightarrow$ Yes
Which of the following numbers is not divisible by 3?
If the sum of the digits of a whole number is divisible by 3, then the number itself is divisible by 3.
$7,310$ ($\text{7+3+1+0} = 11$, and $\rightarrow$/3 = No) No No
$1,722$ ($\text{1+7+2+2} = 12$, and $\rightarrow$/3 = Yes) Yes Yes
$6,411$ ($\text{6+4+1+1} = 12$, and $\rightarrow$/3 = Yes) Yes Yes
$7,515$ ($\text{7+5+1+5} = 18$, and $\rightarrow$/3 = Yes) Yes Yes
Which of the following numbers is not divisible by 4?
If the number represented by the last two digits of a whole number is divisible by 4, then the number itself is divisible by 4.
$5,442$ ($5,442\div4 = 10.5$) $\rightarrow$ No
$9,244$ ($9,244\div4 = 11$) $\rightarrow$ Yes
$5,172$ ($5,172\div4 = 18$) $\rightarrow$ Yes
$6,036$ ($6,036\div4 = 9$) $\rightarrow$ Yes
Which of the following numbers is not divisible by 5?
If a whole number ends in a zero or a 5, then the number is divisible by 5.
$2,039 \rightarrow \text{No}$
$5,955 \rightarrow \text{Yes}$
$2,635 \rightarrow \text{Yes}$
$8,025 \rightarrow \text{Yes}$
Which of the following numbers is not divisible by 6?
If a whole number is divisible by 2 and by 3, then it is divisible by 6.
$4,114$ ($4,114$ is even) $\rightarrow$ Yes
$4,114$ ($\text{4+1+1+4} = 10$, and $\rightarrow$/3 = No) No No
KaTeX can only parse string typed expression
$3,966$ ($3,966$ is even) $\rightarrow$ Yes
$3,966$ ($\text{3+9+6+6} = 24$, and $\rightarrow$/3 = Yes) Yes Yes
KaTeX can only parse string typed expression
$1,920$ ($1,920$ is even) $\rightarrow$ Yes
$1,920$ ($\text{1+9+2+0} = 12$, and $\rightarrow$/3 = Yes) Yes Yes
KaTeX can only parse string typed expression
$6,726$ ($6,726$ is even) $\rightarrow$ Yes
$6,726$ ($\text{6+7+2+6} = 21$, and $\rightarrow$/3 = Yes) Yes Yes
KaTeX can only parse string typed expression
Which of the following numbers is not divisible by 8?
If the number represented by the last three digits of a whole number is divisible by 8, then the number itself is divisible by 8
$2,181$ ($2,181\div8 = 22.625$) $\rightarrow$ No
$2,288$ ($2,288\div8 = 36$) $\rightarrow$ Yes
$2,664$ ($2,664\div8 = 83$) $\rightarrow$ Yes
$5,200$ ($5,200\div8 = 25$) $\rightarrow$ Yes
Which of the following numbers is not divisible by 9?
If the sum of the digits of a whole number is divisible by 9, then the number itself is divisible by 9.
$5,149$ ($\text{5+1+4+9} = 19$) $\rightarrow$ No
$6,633$ ($\text{6+6+3+3} = 18$) $\rightarrow$ Yes
$2,493$ ($\text{2+4+9+3} = 18$) $\rightarrow$ Yes
$2,241$ ($\text{2+2+4+1} = 9$) $\rightarrow$ Yes