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Just For Fun

Divisibility rules for numbers 7-12

Following on from the previous blog, which showed you a series of tricks you can use to determine if a number is divisible by the numbers 2 to 6, this blog will build on this for the numbers 7, 8, 9, 10, 11 and 12.

Rule for 7

This is a tricky one.

To check if a number is divisible by 7, you need to subtract 2 times the last digit from the rest of the digits. If this result is divisible by 7, then the original number is divisible by 7.

Let’s look at some examples:

364

I’ve split this into colours to make it clearer. We are going to take the red numbers, and subtract 2 × the blue number.

36 – 2 × 4  = 28

28 is divisible by 7, so 364 is divisible by 7.

1013

101 – 2 × 3 = 95

95 is not divisible by 7, so 1,013 is not divisible by 7.

You can see the problem with this rule – if you are testing numbers with 4 digits or more, the calculations become much longer and they are no longer calculations that most people can do in their heads.

Rule for 8

This one is much easier than 7.

If the hundreds digit (the number 3 from the right) is even, then if the number formed by the last two digits is divisible by 8 then so is the number.

If the hundreds digit is odd, then add 4 to the number formed by the last two digits. If this number is divisible by 8 then so is the number.


Let’s do some examples:

Again I am going to colour-code – the hundreds digit is in red, and the last two digits are in blue.

656

The hundreds digit is even, so I just need to see if the last two digits are divisible by 8.

56 ÷ 8 = 7 so this means that 656 is divisible by 8.

1,776

The hundreds digit is odd, so I need to add 4 to the last two digits and see if this is divisible by 8.

76 + 4 = 80

80 ÷ 8 = 10 so this means that 1,776 is divisible by 8.

2,342

The hundreds digit is odd again, so I need to add 4 to the last two digits and see if this is divisible by 8.

42 + 4 = 46

46 ÷ 8 = 5.75 so this means that 2,342 is not divisible by 8.

The great thing about this rule is that it doesn’t matter how big the number is, because you only need to use the last 3 digits. So long as you know your 8 times table up to 100, you’ll be fine! Let me show you by trying it for an absurdly large number:

567,890,228

The hundreds digit is even, so I just need to see if the last two digits are divisible by 8.

28 ÷ 8 = 3.5 so this means that 567,890,228 is not divisible by 8.

Rule for 9

This rule is very similar to the rule for 3 in the previous blog.

To test that any number is divisible by 9, add up the digits of the number. If that resulting number is divisible by 9, then the original number is divisible by 9.

For example, if I wanted to know if 886 is divisible by 9, I would add up the digits:

8 + 8 + 6 = 22.

22 is not in the 9 times table, so 886 is not divisible by 9.

Here are two more examples:

189,342

1 + 8 + 9 + 3 + 4 + 2 = 27

27 ÷ 9 = 3, so 189,342 is divisible by 9.

8,921,286

8 + 9 + 2 + 1 + 2 + 8 + 6 = 36

36 ÷ 9 = 4, so 8,921,286 is also divisible by 9.

Rule for 10

This is probably the most obvious one of all and hopefully one you already knew.

Any number that ends in 0 is divisible by 10.

So, 125,090 is divisible by 10 because it ends in a 0.
But, 98,145 is not divisible by 10 because it does not end in 0.

Rule for 11

To check a number is divisible by 11, we form the “alternating sum” of the digits. If this result is divisible by 11, then the number is divisible by 11.

The alternating sum means instead of adding the digits together, you take the first digit, subtract the second, add the third, subtract the fourth, add the fifth, and so on…

So for example, if I wanted to test the number 759:

7 – 5 + 9 = 11

11 ÷ 11 = 1, so that means that 759 is divisible by 11.

91,839

9 – 1 + 8 – 3 + 9 = 22

22 ÷ 11 = 2, so that means that 92,839 is divisible by 11.

108,456

1 – 0 + 8 – 4 + 5 – 6 = 4

4 is clearly not divisible by 11, so 108,456 is not divisible by 11.

This will sometimes yield negative results, like this next example:

18,293

1 – 8 + 2 – 9 + 3 = -11

-11 ÷ 11 = -1, so that means that 18,293 is divisible by 11.

Rule for 12

If a number is divisible by 12, it must be divisible by both 3 and 4. So to check if a number is divisible by 12, we just apply the rules for 3 and 4 from the previous blog.

A quick reminder of how to test for 3:

To test that any number is divisible by 3, add up the digits of the number. If that resulting number is divisible by 3, then the original number is divisible by 3.

And for 4:

To check if a number is divisible by 4, you take the last two digits of the number, and if that number is a multiple of 4 then so is your original number.

Now let’s use both of them to test for 12:

For example, if I wanted to know if 744 is divisible by 12, I’d first check for 3:

7 + 4 + 4 = 15
15 ÷ 3 = 5. This is a whole number.
So 744 is divisible by 3.

Now I’d check for 4:

The last two digits are 44. This is divisible by 4.
So 744 is also divisible by 4.

Because 744 is divisible by both 3 and 4, it is divisible by 12.

Another example:

4,074

Test for 3:

4 + 0 + 7 + 4 = 15
15 ÷ 3 = 5
So it is divisible by 3. Now let’s try 4:

The last two digits are 74. 74 ÷ 4 = 18.5 so it is not divisible by 4.

Although 4,074 is divisible by 3, it is not divisible by 4.

So 4,074 is not divisible by 12.

If you need some extra help with arithmetic or any aspect of mathematics, book a free taster session.

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