# Try this: The sieve of Eratosthenes

You will need

* A grid containing each of the numbers from one to one hundred. You can download one here
* Several colours of pencil or texta

What to do

In this activity, you will find all the prime numbers under 100. A prime number can’t be divided evenly by any whole number except one and itself. For example, three is a prime number. Six can be divided evenly by two and three, so it isn’t a prime.

1. Start out by circling 2. Two is a prime number because it can only be divided by one and two. However, any other number that is divisible by two isn’t prime. Using one colour of pencil or texta, cross out every other even number. You should find they form neat rows.
2. Now circle 3. Three is also a prime number, but any other multiple of three is not, so cross out all the multiples of three with a different coloured texta. You’ll notice these run in diagonal lines across the page.
3. The next number that isn’t crossed out is 5. Five is a prime, so you can circle it, and then cross out all the multiples of five with a third colour. These should be easy to find – the only ones you haven’t yet crossed out should be directly below the 5.
4. Find the next smallest number that isn’t circled or crossed out. Circle it and then cross out all of its multiples. Keep repeating this process until all the numbers on the grid are crossed out or circled.
5. Through this process you have crossed out all the numbers that aren’t prime. The circled numbers are all the prime numbers under 100.

What’s happening?

This method of finding prime numbers is thousands of years old, and was probably invented by an ancient Greek called Eratosthenes in the third century BCE. It’s a good way to make a list of prime numbers relatively quickly.

When you’re using this method, you might find you stop crossing out numbers a lot earlier than you thought. If you start with 100 numbers, you shouldn’t need to cross out any numbers after you cross out all the multiples of seven. That’s because those multiples have another factor that is smaller.

If your grid was larger, you’d need to cross out multiples of larger primes. If your grid went up to 121, you’d need to cross out a multiple of 11, and if it went up to 400, you’d need to cross out multiples of every prime smaller than 20. With each new prime number you come to, the first box you’ll cross off is that prime number multiplied by itself.
Applications

This method is good for finding all the prime numbers, but in a lot of cases, not all of them are needed. For example, prime numbers for a code are often really big, with hundreds of digits. There isn’t enough space on all the hard drives on the Earth to keep a list of the primes that large, so this method can’t be used. If a list of all the big primes could be made, they wouldn’t be very good for making codes!

To find big prime numbers for codes, computers start with a big random number, and then check to see if it’s prime. If it isn’t, they try a different number. This way, there’s a very good chance that it will find a prime no one has ever found before.

What you will need.

Circle each prime number, and then cross out their multiples.

Use different colours for each prime to show some interesting patterns.