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Nov27

Prime Numbers: How to Find Them with the Sieve of Eratosthenes

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In today’s post, we’re going to learn how to find prime numbers using the Sieve of Eratosthenes.

A prime number is one which is only divisible by 1 and itself. It’s as simple as that; the downside is that there’s no mathematical formula to make sure whether a number is prime or not.

Think of a high number like 191,587. We don’t have the formula to determine whether or not it’s prime!

We have to find out if it has any divisors, and is therefore composite. In this case, we would discard it as a prime number.

It’s easy to check if the first few prime numbers (2, 3, 5, 7, 11) have divisors using the help of divisibility criteria. But it’s not so easy for larger numbers.

Imagine having to check all the divisors of such a large number! It would be crazy!

The Sieve of Eratosthenes

The Greek mathematician Eratosthenes (3rd-century B.C.E) designed a quick way to find all the prime numbers. It’s a process called the Sieve of Eratosthenes. We’re going to see how it works by finding all the prime numbers between 1 and 100.

The idea is to find numbers in the table that are multiples of a number and therefore composite, to discard them as prime. The numbers that are left will be prime numbers.

The Sieve of Eratosthenes stops when the square of the number we are testing is greater than the last number on the grid (in our case 100).

Since 11 = 121 and 121>100, when we get to the number 11, we can stop looking.

Prime numbers between 1 and 100 with the Sieve of Eratosthenes

We start by placing the numbers from 1 to 100 in a table like this. This way it’s very easy to see the patterns that the multiples of each number make. We highlight the 1, which is not a prime number.

prime numbers

  • First, we look for the multiples of 2 and highlight them (leaving the 2, since we know it only has divisors of 1 and 2 and is therefore prime). All the highlighted numbers will be composite. Have you seen the lovely pattern that the even numbers make?

https://www.smartickmethod.com/blog/wp-content/uploads/Tabla100.jpg

  • Now, from the numbers that are left, we look for the multiples of 3 and highlight them (except for 3, since it’s prime). An easy way to do it is by counting in threes. We get another interesting pattern when we’re done.

prime numbers

  • Now it’s time to look for the multiples of 5. We don’t need to look for the multiples of 4, because all the multiples of 4 are also multiples of 2, so we’ve already highlighted them. It’s easy to find the multiples of 5, they all end in either 0 or 5. We don’t highlight the 5, because it’s prime.

prime numbers

  • Let’s move on to the multiples of 7 (6 = 2 x 3 and we’ve already found the multiples of 2 and 3). We don’t highlight the 7 since it’s prime.

prime numbers

Do we have to look for the multiples of 8, 9 and 10? Since these numbers are composite and multiples of numbers that we’ve already looked for, we can move on to the number 11. We’ve already established that we stop at the number 11, so that means we’ve finished!

List of prime numbers between 1 and 100

We can, therefore, determine that the numbers that we haven’t highlighted are all prime numbers. So now we have the list of prime numbers between 1 and 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

See how easy it is to look for prime numbers with this method?

If you found that fun, take a look at the book “The Number Devil. I highly recommend it if you want to learn a lot about math in a very entertaining way.

Finally, here you have an image including all the steps that we saw in the example, so you can see them all together.

prime numbers

Have you enjoyed this post? Did you already know about the Sieve of Eratosthenes? I dare you to try it yourself by downloading and printing this table of numbers from 1 to 100. You can do what we’ve done in this post by crossing out the numbers that aren’t prime.

If you want to keep learning and practicing primary mathematics, log in to Smartick and try our learning method.

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7 Comments

  • Joshua DinkinsApr 05 2019, 10:58 AM

    Amazingly fun without all the hassle of challenging division to see if prime is an existence. At first, I’d always wonder why it was so difficult to find this kind of number at its efficient levels, but now the Sieve of Eratosthenes has been a complete advisory to my knowledge. Now I’m moving forward to the next sequential element of prime numbers; Goldbach’s Conjectures. Thank You.

    Reply
    • SmartickApr 08 2019, 1:05 AM

      Thank you very much for your reply to this post, Joshua!

      Reply
  • ramasaiFeb 12 2019, 10:08 PM

    When I first looked it in theory, it was a little bit confusing, now it just took me seconds to understand that. Thank you, keep up the good work.

    Reply
  • Lilly hareFeb 06 2019, 1:08 PM

    Well done

    Reply
  • AarthiOct 13 2018, 8:10 AM

    Nice

    Reply