## 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.

239879**8** is divisible by 2 because 8 is even.

672364**3** 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.

17**16** is divisible by 4, because 16 is a multiple of four.

17**18** 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.

872053**5** is a multiple of 5.

872053**4** 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.

10**416**. 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.)

10**414**. 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.