Just For Fun

# Divisibility rules for numbers 2-6

In this blog I will show you how you can easily check any number is divisible by the numbers 2-6. So simple that should your mental arithmetic be strong enough, you will be able to just take a glance at the number and know immediately whether it is divisible by these numbers.

Firstly, what do I mean by divisible?

Number A is divisible by Number B if you can divide Number A by Number B and end up with a whole number.

For example, 162 is divisible by 3 because 162 ÷ 3 = 54 (which is a whole number), but it is not divisible by 4 because 162 ÷ 4 = 40.5 (which is not a whole number).

Note: Saying “Number A is divisible by Number B” is exactly the same thing as saying “Number A is a multiple of Number B”.

Now that we understand what divisibility means, let me show you the rules.

## Rule for 2

This one is very simple. You may have already known this one before reading this blog. Any even number is divisible by 2. And an even number is any number that ends in an even number!

So any number that ends in 0, 2, 4, 6 or 8 must be divisible by 2.

So for example:

123,456 is divisible by 2 because it ends in 6
77,890 is also divisible by 2 because it ends in 0
56,453 is NOT divisible by 2 because it ends in 3

## Rule for 3

This one is less obvious.

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.

So, for example, if I wanted to know if 327 is divisible by 3, I would add up the digits:

Add up the digits: 3 + 2 + 7 = 12.

12 ÷ 3 = 4. This is a whole number.

Because 12 is divisible by 3, 327 is also divisible by 3.

Another example – is 470,124 divisible by 3?

Add up the digits: 4 + 7 + 0 + 1 + 2 + 4 = 18.

18 ÷ 3 = 6. So 470,124 is also divisible by 3.

Another example – is 422,411 divisible by 3?

4 + 2 + 2+ 4 + 1 + 1 = 14

14 is not divisible by 3, so 422,411 is not divisible by 3.

You’ll notice for this one you need to be quick at adding single digits up, you need to know your 3 times table pretty well to be able to spot them quickly and clearly the more digits the number has, the longer the process will take.

## Rule 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.

For example, if I wanted to know if 336 is divisible by 4, I would take the last two digits (36). Because 36 ÷ 4 = 9, 336 is divisible by 4.

Here are two more examples:

521,460

Last two digits = 60
60 ÷ 4 = 15
So 521,460 is divisible by 4.

677,134

Last two digits = 34
34 ÷ 4 = 8.5
So 677,134 is not divisible by 4.

The benefit of this trick is that you only need to know your 4 times table up to the last 2-digit number (96). And once you know these, you will be able to tell very quickly that really large numbers are divisible by 4.

For example, 455,209,516 is a very large number… but straight away I can see that because it ends in 16 and 16 is a multiple of 4 that it’s divisible by 4! So unlike the rule for 3, the number of digits the number has makes no difference.

## Rule for 5

Another very easy one. You will know from your 5 times table that each number in the times table ends in either 5 or 0.

So, a number is divisible by 5 if it ends in 5 or 0.

For example:

445 is divisible by 5 because it ends in 5
570,450 is divisible by 5 because it ends in 0
769,923 is NOT divisible by 5 because it ends in 3

## Rule for 6

If a number is divisible by 6, it must be divisible by both 2 and 3. So to check if a number is divisible by 6, we just apply the rules for 2 and 3 from above.

So, a number is divisible by 6 if it ends in an even number AND its digits add to a multiple of 3.

For example, let’s test 624.

It ends in 4, so it is divisible by 2.

6 + 2 + 4 = 12, and 12 is a multiple of 3. So it is also divisible by 3.

Because it is divisible by both 2 and 3, 624 is divisible by 6.

Now let’s test 716.

It ends in 6, so it is divisible by 2.

7 + 1 + 6 = 14, and 14 is not a multiple of 3. So it is not divisible by 3.

Therefore 716 is NOT divisible by 6.

Now that we’ve seen all the rules, let’s use them all in a couple of examples.

Let’s say I had the number 456,780 and I wanted to know if it was divisible by 2, 3, 4, 5 and 6.

Is it divisible by 2? It ends in 0 so YES.
Is it divisible by 3? 4 + 5 + 6 + 7 + 8 + 0 = 30. 30 ÷ 3 = 10 so YES.
Is it divisible by 4? Last two digits = 80. 80 ÷ 4 = 20 so YES.
Is it divisible by 5? It ends in 0 so YES.
Is it divisible by 6? We have already tested 2 and 3, so because it is divisible by 2 and 3, the answer is YES.

So 456,780 is divisible by all numbers from 2-6! And this can be worked out very quickly.

## A real-world example

A real-world application of this could be that I have 1,334,642 people, and I want to split them evenly into groups of either 2, 3, 4, 5 or 6. By using the divisibility rules, I can work out how many of these are possible.

Is it divisible by 2? It ends in 2 so YES.
Is it divisible by 3? 1 + 3 + 3 + 4 + 6 + 4 + 2 = 23. 23 is not a multiple of 3 so NO.
Is it divisible by 4? Last two digits = 42. 42 is not a multiple of 4 so NO.
Is it divisible by 5? It ends in 2 so NO.
Is it divisible by 6? It is divisible by 2 but not 3, so NO.

So I would know that I can only split that group of people into groups of 2. None of the other sizes of group would work.

So there it is, a quick way to test divisibility for the numbers 2-6. These are great tricks to impress your friends and family, and more importantly, to improve your mental arithmetic.

Look out for a future blog which will build on this and introduce divisibility rules for more numbers!

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

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