# Introducing prime numbers

A prime number is any whole number which cannot be shared or divided into equal parts.

#### Numbers that are not prime have factors

For example, the number six is not a prime number. You can share six biscuits equally between three children (they get two each) or between two children, who get three each. You can also, of course, share six biscuits between six children (one each) or give them all to one child – but you can do that kind of thing with any number. With six, however, as well as sharing between six children, you can ALSO share between some other numbers of children. Two and three are known as ‘factors‘ of six.

#### Prime numbers do not have factors

But if you have five biscuits, the only way you can share them out equally is if you have five children (who get one each) or just one lucky child, who gets all five. There is no other way of sharing the biscuits, unless of course you break them, but that would take us into the realm of fractions. Prime numbers and factors deal only in whole, unbroken numbers.

That’s all there is to it.

#### The Sieve of Eratosthenes

If you want to find out which numbers are prime, a clever way of doing so was invented by an Ancient Greek called Eratosthenes (pronounced ‘error – TOSS – the – knees’). He suggested drawing a grid of numbers from 1 to 100, in a box of 10×10 squares. Like this:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Then, because prime numbers are those which cannot be divided by any other numbers (other than 1) go through and cross out all the numbers that CAN be divided by others. That might sound a bit daunting, but start with numbers that divide by two. The first number that divides by two (other than 2, which doesn’t divide by anything else) is 4 (2×2) so cross out the 4. The next number is 6, then 8, and so on, right the way up to 100. This is easy, because you cross out every other number, and will end up with a sort of pattern of stripes:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Next, cross out numbers that divide by 3. This is a little more tricky, since some of them have already been crossed out – such as 6 and 12, which also divide by 2. But keep counting (or check on your calculator) and you’ll find that you also need to cross out 9, 15, 21, 27, 33, 39, and so on right the way up to 99.

Next should be numbers that divide by 4. However you don’t actually need to cross anything out here. Why not? Well four itself divides by 2 (ie 2 is a factor of 4). That means that any number which divides by four can ALSO be divided by 2. Look at the grid if you’re not sure. 2×4=8, 3×4=12, 4×4=16, 5×4=20… and so on. Anything which divides by 4 has already been crossed out.

Next cross out the numbers that divide by 5. You will find that some of them have already gone, but there are still a few left: 25, 35, 55, 65, 85, and 95. Cross these out too.

You don’t need to worry about numbers that divide by 6. They have already been dealt with – twice, in fact! 2 and 3 are both factors of six, so any number which can be divided into six parts can also be divided into two or three.

7, however, does not have any factors, so it’s another prime number. Look at numbers which divide by 7 left on the grid. Most of them have gone, but there are still a few remining: 49, 77, and 91.

Do you need to continue? Well, you can try if you wish, but there aren’t any more numbers to cross out. Any number which divides by 8 also divides by 2, so they’re already gone. Any number which divides by 9 also divides by 3, so they’re gone too. 10 divides by 2 as well.

What about 11? This is a bit more complicated, since 11 is another prime number. But you crossed out 22 (2×11) when you dealt with the numbers that divided by 2, and 33 (3×11) when you dealt with the threes, 44 (4×11) with the 2s, 55 (5×11) with the 5s, and so on. Right up to 99 which is 9×11, and was dealt with when you crossed out the numbers that divided by 3. The first number that can be divided by 11 which has not been crossed out is 121, which is 11×11 – but that’s not on this grid, as we only went up to 100.

The final result is the Sieve of Eratosthenes, showing all prime numbers less than 100:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

You can continue discovering bigger prime numbers by drawing bigger grids – up to 200, or 500, or any number you like, but it gets a bit tedious by the time you have to work out numbers that divide by 59 or 67… although if your child is interested, a calculator makes this fairly painless.

To introduce other basic maths, see:

### Home Education Book

Home Education - What? Why? How?: A Guide for Beginners 