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# Divisibility tests for 2, 3, 4, 5, 6, 9, 10

Worked example of basic divisibility tests. Created by Sal Khan.

## Want to join the conversation?

• What about divisibility by 7 and 8? Guess there's no easy way to test that without a calculator ;) •   Actually, divisibility by 7 & 8 is quite easy once you get the hang of it.

First, I will talk about divisibility by 8, since it is easier. In order to test this, you only must check to see whether the last three digits of the number are divisible by 8. If they are, then the entire number is divisible by 8 too.

Example 1: Is the number 8347475537272 divisible by 8?
Answer 1: Yes, because the last 3 digits, 272, are divisible by 8.

Example 2: Is the number 314159265358979323846 divisible by 8?
Answer 2: No, because the last 3 digits, 846, are not divisible by 8.

Next, divisibility by 7. This one is a little weird but it really is quite simple after you practice it a couple of times. In order to test this, you must take the last digit of the number you’re testing and double it. Then, subtract this number from the rest of the remaining digits. If this new number is either 0 or if it’s a number that’s divisible by 7, then then original number is divisible by seven. (You may have to repeat this a couple of times if the divisibility of the resulting number is not immediately obvious).

Example 1: Is the number 364 divisible by 7?
Answer 1: Yes: Double the 4 to get 8. Subtract 8 from 36 to get 28. Since 28 is divisible by 7, we can now say for certain that 364 is also divisible by 7.

Example 2: Is the number 8256 divisible by 7?
Answer 2: No, Double 6 to get 12. Subtract 12 from 825 to get 813. 813 is slightly too large to tell whether it is divisible by 7 so we must repeat the process. Double 3 to get 6. Subtract 6 from 81 to get 75. Since 75 is not divisible by 7, neither is 813 or 8256. Therefore, 8256 is not divisible by 7.

If I wasn't clear with my explanation or if you need any more help, just ask. I hope that helped. :)

P.S. This edit was made in response to a.ortalda's great question about divisibility by 7. It is as follows:

What about the number used in this video? I tried to test the divisibility by 7 of 2799588, but at a certain point I have 27994 - (2*2) = 27990. How can I continue?

For this special case, you can just drop the zero from 27990 to 2799 and continue from there. This works because, in essence, you are dividing by 10. Since 10 equals 5*2 (neither of which are 7), it should not influence the result.

To finish the problem, 279 - 9*2 = 261. 26 - 1*2 = 24. Since 24 is not divisible by 7, neither is 2799588.
• Divisibility by 11:
How to check a number is divisible by 11?
It is very simple, check the number, if the difference of the sum of digits at odd places and the sum of its digits at even places, is either 0 or divisible by 11, then clearly the number is divisible by 11.

Examples:
Is the number 2547039 divisible by 11?
First find the difference between sum of its digits at odd and even places.
(Sum of digits at odd places) - (Sum of digits at even places)
= (9 + 0 + 4 + 2) - (3 + 7 + 5)
= 15 - 15 = 0
The number is 0, so the number 2547039 is divisible by 11.

Is the number 13165648 divisible by 11?
(Sum of digits at odd places) - (Sum of digits at even places)
= (8 + 6 + 6 + 3) - (4 + 5 + 1 + 1)
= 23 - 11 = 12
The number is 12, so the number 13165648 is not divisible by 11.
Divisibility by 11:
How to check a number is divisible by 11?
It is very simple, check the number, if the difference of the sum of digits at odd places and the sum of its digits at even places, is either 0 or divisible by 11, then clearly the number is divisible by 11.

Examples:
Is the number 2547039 divisible by 11?
First find the difference between sum of its digits at odd and even places.
(Sum of digits at odd places) - (Sum of digits at even places)
= (9 + 0 + 4 + 2) - (3 + 7 + 5)
= 15 - 15 = 0
The number is 0, so the number 2547039 is divisible by 11.

Is the number 13165648 divisible by 11?
(Sum of digits at odd places) - (Sum of digits at even places)
= (8 + 6 + 6 + 3) - (4 + 5 + 1 + 1)
= 23 - 11 = 12
The number is 12, so the number 13165648 is not divisible by 11. • is there a way to find the multiple of 11 when it's a three digit number? is it possible? • • Dividing by 7 (2 Tests)

Take the last digit in a number.
Double and subtract the last digit in your number from the rest of the digits.
Repeat the process for larger numbers.
Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

NEXT TEST
Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary
If the sum is divisible by 7 - so is your number.
Example: Is 2016 divisible by 7?
6(1) + 1(3) + 0(2) + 2(6) = 21
21 is divisible by 7 and we can now say that 2016 is also divisible by 7.

Dividing by 8

This one's not as easy, if the last 3 digits are divisible by 8, so is the entire number.
Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008.
• • is there anything as partial divisibility and if there is where can we find its usage? like 90 being divisible by 4 partially because 90/4=22.5 • • • • It's not simple like some of the others, but there is one. Here it is:
To determine if an integer is divisible by 13, remove the last digit, multiply it by 9, and subtract this from the remaining digits. Continue this until you reach a number that you know is divisible or not divisible by 13. If it is divisible by 13, then the original number is divisible by 13, but if it is not divisible by 13, the original number is not. For example, 312 is divisible by 13, because multiplying the last digit by 9 and subtracting it from the rest of the number yields 31 - 9(2) = 13, which is divisible by 13, but 867 is not divisible by 13, because multiplying the last digit by 9 and subtracting it from the rest of the number yields 86 - 9(7) = 23, which is not divisible by 13. 