# Divisibility and PrimesDivisibility Rules

There are a few different rules that can make it surprisingly easy to check if a number is divisible by another. In this section we will have a look at some of them…

## Divisibility by 2 and 5

Every number is divisible by 1. To determine if a number is divisible by 2, we simply have to check if it’s even: any number that ends in 0, 2, 4, 6, or 8 is divisible by 2.

To see if a number is divisible by 5, we similarly just have to check that its last digit is 0 or 5:

The reason why these rules for 2 and 5 are so simple has to do with our number system. The base of our number system is 10, which means that every digit in a number is worth 10 times as much as the next one to the right. If we take the number 6382 as an example,

6 | 3 | 8 | 2 |

=6000 | =300 | =80 | =2 |

Now we can separate the last digit of a number from all its other digits:

abcd | = | abc × 10 | + | d |

6382 | = | 638 × 10 | + | 2 |

Both 2 and 5 are factors of 10, so they will **abc × 10**, no matter what the values of **a**, **b** and **c** are. Therefore we only have to check the last digit: if **d** is divisible by 2 then **d** is divisible by 5 then the whole number is divisible by 5.

The easiest is the divisibility rule for 10: we just need to check if the

## Divisibility by 4 and 8

Unfortunately 4 doesn’t divide 10, so we can’t just look at the last number – but 4 *does* divide 100, so we just have to slightly modify our rule from above. Now we write **ab****cd** = **ab × 100** + **cd**. We know that 4 will always divide **ab × 100**, so we have to look at the last

For example, **24** is divisible by 4 so **2735****24** **18** is not divisible by 4 so **1947****18**

The divisibility rules for 8 get even more difficult, because 100 is not divisible by 8. Instead we have to go up to

For example, **120** is divisible by 8 so **271****120** is also divisible by 8.

## Divisibility by 3 and 9

The divisibility rule for 3 is rather more difficult. 3 doesn’t divide 10, and it also doesn’t divide 100, or 1000, or any larger power of 10. Simply looking at the last few digits of a number isn’t going to work.

Instead we need to use the **digit sum** of a number, which is simply the sum of all its individual digits. For example, the digit sum of

Here we’ve highlighted all numbers which are multiples of three. You can see that their digit sums are always

So to determine if any number is divisible by 3, you just have to calculate its digit sum, and check if the result is also divisible by 3.

Next, let’s look at multiples of 9:

It seems that all the numbers divisible by 9 have a digit sum which is

Of course, these curious patterns for numbers divisible by 3 and 9 must have some reason – and like before it has to do with our base 10 numbers system. As we saw, writing the number **6****3****8****4** really means

**6 × 1000** + **3 × 100** + **8 × 10** + **4**.

We can split up each of these products into two parts:

**6 × 999 + 6** + **3 × 99 + 3** + **8 × 9 + 8** + **4**.

Of course, **9**, **99**, **999**, and so on are always divisible by 3 (or by 9). All that remains is to check that what’s left over is also divisible by 3 (or 9):

**6** + **3** + **8** + **4**

This just happens to be the digit sum! So if the

## Divisibility by 6

We’ve still skipped number 6 – but we’ve already done all the hard work. Remember that 6 = 2 × 3.

To check if a number is divisible by 6 we just have to check that it is divisible by 2 *any* number that is the product of two others. More on that later…