**Divisibility Rules** *KS3 Revision*

## What you need to know

**Things to remember:**

- Some rules need to be performed multiple times.
- Divisibility rules tell you if a number can be divided, not the answer.

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__Dividing by 1__

Any whole number is divisible by 1.

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__Dividing by 2__

If we look at the first 10 numbers in the 2 times table:

2 4 6 8 10 12 14 16 18 20

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We can see that they end in a 2, 4, 6, 8 or 0. So, if you see a number ending in one of these numbers, we can always divide it by 2.

__Dividing by 3__

If we can divide the sum of the digits by 3, then it is itself divisible by 3.

*Is 246 divisible by 3?*

2+4+6=12

12\div3=4

246 is divisible by **3**!

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You can do it twice to be certain.

*Is 36570 divisible by 3?*

3+6+5+7+0=21

2+1=3

3\div3=1

36570 is divisible by 3.

__Dividing by 4__

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100\div4=25

1000\div4=250

10000\div4=2500

etc.

Any multiple of a power of 10 that is 100 or more is divisible by 4, so we just ned to look at the last two digits. If the last two digits are divisible by 4, the whole number is divisible by 4.

*Is 256 divisible by 4?*

56\div4=14

256 is divisible by 4

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__Dividing by 5__

If we look at the first 10 numbers in the 5 times table:

5 10 15 20 25 30 35 40 45 50

We can see that they end in 5 or 0. So, if you see a number ending in one of these, we can always divide it by 5. We can divide the following by 5:

40 120 165 825 105 2005

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__Dividing by 6__

If we think about 6 as 2\times3 , when we divide by 6 we are actually dividing by 2 and 3. So, if a number satisfies the rules for dividing by 2 and 3, we can divide it by 6

Is 756 divisible by 6?

756 ends in a 6, so is divisible by 2.

7+5+6=18

18\div3=6

756 is divisible by 3.

756 is divisible by 2 and 3, so is divisible by 6.

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__Dividing by 7__

To check for the divisibility by 7, we do it in three steps.

**Step 1: **Double the last digit.

**Step 2:** Subtract it from the remaining digits.

**Step 3:** See if the number in Step 2 is divisible by 7.

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*Is 1673 divisible by 7?*

__Step 1__

3\times2=6

__Step 2__

167-6=161

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__Step 3__

161\div7=23

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1673 is divisible by 7

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__Dividing by 8__

100\div8=12.5

1000\div8=125

10000\div4=1250

Unlike with the 4 times table, only powers of 10 that are 1000 or bigger are divisible by 8, so we have to look at the last 3 digits to see if they are divisible by 8.

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*Is 108336 divisible by 8?*

336\div8=42

108336 is divisible by 8.

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__Dividing by 9__

If the sum of the digits in the number are divisible by 9, the whole number is divisible by 9

*Is 122814 divisible by 9?*

1+2+2+8+1+4=18

18\div9=2

122814 is divisible by 9

We can do this multiple times.

*Is 893691 divisible by 9?*

8+9+3+6+9+1=36

3+6=9

9\div9=1

893691 is divisible by 9.

* *

__Dividing by 10__

If we look at the first 10 numbers in the 10 times table:

10 20 30 40 50 60 70 80 90 100

We can see that they always end in a 0. So, if you see a number ending in 0, we can divide it by 10.

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__Dividing by 11__

If we add and subtract the digits in an alternating patter and the result is divisible by 11, the whole number is divisible by 11.

*Is 260227 divisible by 11?*

2-6+0-2+2-7= -11

-11\div11=-1

260227 is divisible by 11.

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__Dividing by 12__

If we think about 12 as 3\times44 , when we divide by 12 we are actually dividing by 3 and 4. So, if a number satisfies the rules for dividing by 3 and 4, we can divide it by 12.

*Is 161616 divisible by 12?*

* *

1+6+1+6+1+6=21

21\div3=7

161616 is divisible by 3

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16\div4=4

161616 is divisible by 12

161616 is divisible by 3 and 4, so is divisible by 12.

## Example Questions

**Question 1:** Is 1268 divisible by 7?

**Step 1: **Double the last digit.

**Step 2:** Subtract it from the remaining digits.

**Step 3:** See if the number in Step 2 is divisible by 7.

__Step 1__

8\times2=16

__Step 2__

126-16=110

__Step 3__

110\div7=15.\overline{714285}

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Step 3 didn’t give a whole number, so 1268 is not divisible by 7.

**Question 2:** Is 14816703 divisible by 11?

14816703

1-4+8-1+6-7+0-3=0

0\div11=0

14816703 is divisible by 11.