 ## 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.

Dividing by 1

Any whole number is divisible by 1.

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

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!

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

$$100\div4=25$$

$$1000\div4=250$$

$$10000\div4=2500$$

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

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

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.

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.

Is 1673 divisible by 7?

Step 1

$$3\times2=6$$

Step 2

$$167-6=161$$

Step 3

$$161\div7=23$$

1673 is divisible by 7

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.

Is 108336 divisible by 8?

$$336\div8=42$$

108336 is divisible by 8.

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.

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.

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

$$16\div4=4$$

161616 is divisible by 12

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

## Example Questions

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

$$110\div7=15.\overline{714285}$$

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

14816703

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

$$0\div11=0$$

14816703 is divisible by 11.