Divisibility tricks

2 05 2010

Is 6783043998 divisible by 3 without getting a remainder? Is it divisible by 7, 9 or 11? You don’t have to resort to the calculator to work this out, there are some great tricks for testing the divisibility of numbers.

Here’s the rules for checking divisibility by numbers between 1 and 10:

Divisibility by 2

This one is nice and straight forward. All even numbers are divisible by 2 so if the last digit of your number is 0, 2, 4, 6, 8 then it is divisible by 2 without getting a remainder.

Divisibility by 3

There’s a really neat trick that you can use to check for divisibility by 3. Just add together all the digits in your number and see if the result is divisible by 3. If it is then the original number is divisible by 3 without getting a remainder.

Is 543 divisible by 3?

5 + 4 + 3 = 12

12 is divisible by 3 so 543 is also divisible by 3

If the result you get from adding together the digits is a big number where it isn’t obvious straight away if you can divide it by 3 without getting a remainder then you can take the result and do it again.

Is 6783043998 divisible by 3?

6 + 7 + 8 + 3 + 0 + 4 + 3 + 9 + 9 + 8 = 57

5 + 7 = 12

12 is divisible by 3 so 6783043998 is also divisible by 3

Divisibility by 4

To check a number for divisibility by 4 you just have to look at the last two digits in the number, no matter how large it is. If you take just the last two digits in a number and that number is divisible by 4 then the original number will be divisible by 4.

Is 912 divisible by 4?

Is divisible by 4 so 912 is also divisibly by 4.

Is 1039 divisible by 4?

39 is not divisible by 4 so 1039 is also not divisible by 4. (1040 would be)

Divisibility by 5

Nice and easy one. If your number ends in 0 or 5 it will be divisible by 5.

Divisibility by 6

If a number is divisible by 2 and by 3 then it will be divisible by 6. Just do the checks for divisibility by 2 and 3.

Divisibility by 7

The method for this one is quite long-winded and it can be quicker to check using short division. Nonetheless, the method is interesting seemingly a little mysterious…

Is 21987 divisible by 7?

You start by taking off the last digit in the number and then doubling it:

7 x 2 = 14

You then subtract this from the remaining digits:

2198 – 14 = 2184

Now you keep doing the same thing until you get to a point where you can see if you have a multiple of 7 or not:

4 x 2 = 8

218 – 8 = 210

0 x 2 = 0

21 – 0 = 21

21 is a multiple of 7, (3 x 7) so 21987 is divisible by 7.

Divisibility by 8

There are two rules to remember for this one:

If the hundreds digit is even look at the last two digits. If the last two digits are divisible by 8 then your original number will be divisible by 8.

Is 345232 divisible by 8?

The hundreds digit is 2 so it is even. The last two digits are 32 which is divisible by 8 so 345232 is divisible by 8.

If the hundreds digit is odd look at the last two digits then add four. If this number is divisible by 8 then your original number will be divisible by 8.

Is 9084344 divisible by 8?

The hundreds digit is 3 so it is odd. 44 + 4 = 48 which is divisible by 8 so 9084344 is divisible by 8.

Divisibility by 9

This one is very similar as the rule for checking for divisibility by 3. Add together all the digits in the number and if the answer is divisible by 9 then your original number will be divisible by 9.

Is 2880 divisible by 9?

2 + 8 + 8 + 0 = 18

18 is divisible by 9 so 2880 is divisible by 9.

With really big numbers you can take the answer from summing the digits and then sum the digits in that number to check if it is a multiple of 9. You can keep doing this as many times as you need to similarly with the method for checking for divisibility by 3.

Divisibility by 10

The easiest of them all! If your number ends in 0 then it is divisible by 10.

Final thoughts

In an age where calculators are the norm these tests for divisibility could be seen as nothing more than neat tricks to impress your teachers or your mates. Some of them seem pretty obvious like the tests for divisibility by 2, 5 and 10. Others however seem quite strange and it is not immediately obvious why the rules work like the tests for divisibility by 3, 7 and 9. I think learning about rules like these and trying to figure out how they work is what makes maths fun. If you can figure out why the rules for the tests of divisibility by 3, 7 or 9 work then write your description in the comments section below and share your solution with everyone!

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