Divisibility Rules (1) Worksheet

Instead of actually carrying out a division, it is possible to "test" a number to see if it can be divided exactly by certain numbers.

If a number divides exactly into a number, it is divisible by this number.

24 is divisible by 2, because 24 ÷ 2 is a whole number.

24 is not divisible by 5, because 24 ÷ 5 is not a whole number.

Here is the rule to test divisibility by 11.

Rule for 11

To test a number for divisibility by 11 we must find the sums of alternative digits. This is best illustrated using examples.

Example 1

Determine whether 17281 is divisible by 11.

Method

Underline alternate numbers to get 17226

Add up the underlined digits to get the sum of 1 + 2 + 6 = 9

Add up the non-underlined digits to get the sum of 7 + 2 = 9

If these two digit sums are equal or differ by a multiple of 11, the original number is divisible by 11.

Here they are equal so 17226 is divisible by 11.

Example 2

Determine whether 17281357 is divisible by 11.

Method

Underline alternate numbers to get 17281539

Add up the underlined digits to get the sum of 1 + 2 + 1 + 3 = 7

Add up the non-underlined digits to get the sum of 7 + 8 + 5 + 9 = 29

If these two digit sums are equal or differ by a multiple of 11, the original number is divisible by 11.

Here they differ by 29 - 7 = 22 so 17281539 is divisible by 11.

Is the following number divisible by 11?

4686

Yes

No

Is the following number divisible by 11?

4386

Yes

No

Is the following number divisible by 11?

41110

No

Yes

Is the following number divisible by 11?

42141

No

Yes

Is the following number divisible by 11?

41105

No

Yes

Is the following number divisible by 11?

400015

No

Yes

Is the following number divisible by 11?

73216

No

Yes

Is the following number divisible by 11?

12276

No

Yes

Is the following number divisible by 11?

27093

No

Yes

Is the following number divisible by 11?

27082

No

Yes

Is the following number divisible by 11?

270759

No

Yes

Is the following number divisible by 11?

2703723

No

Yes

Is the following number divisible by 11?

55555

No

Yes

Is the following number divisible by 11?

555555

No

Yes

Is the following number divisible by 11?

1111111

No

Yes

Is the following number divisible by 11?

11111111

No

Yes

Is the following number divisible by 11?

9181909

No

Yes

Is the following number divisible by 11?

918174

No

Yes

Is the following number divisible by 11?

9181744

No

Yes

Is the following number divisible by 11?

100100

No

Yes

ANSWERS

Question 1

Is the following number divisible by 11?

4686

CORRECT ANSWER

Yes

EDDIE SAYS

Do you remember what the trick is? You need to add every other number and if they equal the same amount then it is equal by 11. 4 + 8 = 12 6 + 6 = 12

ANSWERS

Question 2

Is the following number divisible by 11?

4386

CORRECT ANSWER

No

EDDIE SAYS

Try the system again 4+ 8 = 12 3 + 6 = 9 So we know this one cannot be divided by 11.

ANSWERS

Question 3

Is the following number divisible by 11?

41110

CORRECT ANSWER

No

EDDIE SAYS

Remember, we need to add up every other number. 4 + 1 = 5 1 + 1 = 2 So, this cannot be divisible by 11 either.

ANSWERS

Question 4

Is the following number divisible by 11?

42141

CORRECT ANSWER

Yes

EDDIE SAYS

If you did the maths (without a calculator) you should have worked out that this is indeed divisible by 11. Let's see! 4 + 1 + 1 = 6 2 + 4 = 6 As this number is the same, we know that the number must be divisible by 11. Check on a calculator if you don't trust it!

ANSWERS

Question 5

Is the following number divisible by 11?

41105

CORRECT ANSWER

No

EDDIE SAYS

You know what to do now, right? 4 + 1 + 5 = 10 1 + 0 = 1 So, there's our answer! This is not divisible by 11.

ANSWERS

Question 6

Is the following number divisible by 11?

400015

CORRECT ANSWER

Yes

EDDIE SAYS

Makes life much easier don't you think? And, you can do much bigger sums without a calculator! Remember to add every other number: 4 + 0 + 1 = 5 0 + 0 + 5 = 5 As the numbers the same, this means that the number is divisible by 11.

ANSWERS

Question 7

Is the following number divisible by 11?

73216

CORRECT ANSWER

Yes

EDDIE SAYS

Did this one catch you out? Remember there are two parts to this checking process. If in doubt always go back to the introduction and have another look. First, we add up every other number like before: 7 + 2 + 6 = 15 3 + 1 = 4 Then we subtract 4 from 15 15- 4 = 11 This is a multiple of 11, so there we have it!

ANSWERS

Question 8

Is the following number divisible by 11?

12276

CORRECT ANSWER

Yes

EDDIE SAYS

Straight to our system here: 1 + 2 + 6 = 9 2 + 7 = 9 So, we can see straight away this must be divisible by 11.

ANSWERS

Question 9

Is the following number divisible by 11?

27093

CORRECT ANSWER

Yes

EDDIE SAYS

You know the drill now mathematician! 2 + 0 + 3 = 5 7 + 9 = 16 Which looks like it may not be divisible, but now we try our second approach: 16- 5 = 11 So, this is also divisible by 11. No calculator needed!

ANSWERS

Question 10

Is the following number divisible by 11?

27082

CORRECT ANSWER

Yes

EDDIE SAYS

Let's follow the system: 2 + 0 + 2 = 4 7 + 8 = 15 The numbers are not the same, but let's use our second check here. Now we should spot that 15 - 4 = 11 which is a multiple of 11 and therefore, gives us our answer! This number IS divisible by 11.

ANSWERS

Question 11

Is the following number divisible by 11?

270759

CORRECT ANSWER

No

EDDIE SAYS

Let's get adding those alternate numbers... 2 + 0 + 5 = 7 7 + 7 + 9 = 23 So, we should do our second check now, as so far it looks like this is not divisible by 11. 23 - 7 = 16 As this number is not a multiple of 11, this number is not divisible by 11.

ANSWERS

Question 12

Is the following number divisible by 11?

2703723

CORRECT ANSWER

Yes

EDDIE SAYS

Off we go... 2 + 0 + 7 + 3 = 12 7 + 3 + 2 = 12 And... we've found our answer! As the numbers are the same, it must be divisible by 11.

ANSWERS

Question 13

Is the following number divisible by 11?

55555

CORRECT ANSWER

No

EDDIE SAYS

Alternate adding here we come! 5 + 5 + 5 = 15 5 + 5 = 10 When we subtract 10 from 15 we end up with 5, which is NOT a multiple of 11.

ANSWERS

Question 14

Is the following number divisible by 11?

555555

CORRECT ANSWER

Yes

EDDIE SAYS

We shouldn't have to add anything for this one! As all of the numbers are the same, we know that the two sums of the alternate numbers will also be the same, so this number must be divisible by 11. Did you get that logic? Let's test it out just to be sure... 5 + 5 + 5 = 15 5 + 5 + 5 = 15 See, there we go! Just as we thought, it can be divided by 11.

ANSWERS

Question 15

Is the following number divisible by 11?

1111111

CORRECT ANSWER

No

EDDIE SAYS

The clue with this one is checking how many number 1's there are. In this case, there are 7, so we know that adds to make 3 and 4. As these numbers are not the same, and they do not subtract to get a multiple of 11, we can quickly work out that this number is not divisible by 11. Keeping up? Let's try another!

ANSWERS

Question 16

Is the following number divisible by 11?

11111111

CORRECT ANSWER

Yes

EDDIE SAYS

Did you add up how many number ones there were here? (Made my eyes go a bit fully!) There are 8 number ones in this number, so we know that the sum of the alternates must be the same. So we don't need to do the maths, we know this number is divisible by 11. Let's keep moving along, next one, please!

ANSWERS

Question 17

Is the following number divisible by 11?

9181909

CORRECT ANSWER

Yes

EDDIE SAYS

OK - for this one we need to do some alternate maths again. 9 + 8 + 9 + 9 = 35 1 + 1 + 0 = 3 35 - 2 = 33 This is a multiple of 11, so we know that the answer here is YES!

ANSWERS

Question 18

Is the following number divisible by 11?

918174

CORRECT ANSWER

No

EDDIE SAYS

Bit more addition needed: 9 + 8 + 7 = 24 1 + 1 + 4 = 6 24 = 6 = 18 This is not a multiple of 11, so we know that for this, the answer is no!

ANSWERS

Question 19

Is the following number divisible by 11?

9181744

CORRECT ANSWER

Yes

EDDIE SAYS

9 + 8 + 7 + 4 = 28 1 + 1 + 4 = 6 28 - 6 = 22 There's our answer. 22 is a multiple of 11.

ANSWERS

Question 20

Is the following number divisible by 11?

100100

CORRECT ANSWER

Yes

EDDIE SAYS

Last question, let's make it a good one! 1 + 0 + 0 = 1 0 + 1 + 0 = 1 As these numbers are the same, we know that this number must be divisible by 11. Great work, that's another activity completed! Why not try another to test your understanding.