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Math for Kids23 chapters | 325 lessons

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Instructor:
*Nick Rogers*

Prime numbers are important to mathematicians, and there are many ways to find them. The sieve of Eratosthenes is one of the first that was ever developed, which you'll explore here.

Eratosthenes was a Greek scientist who lived nearly 2,000 years ago, studied geography, and made some of the first maps of the entire known world. He is credited with inventing a method for finding prime numbers known as a sieve. Lets review some information about factors and prime numbers that can help us understand his sieve.

Numbers are made up of other numbers called **factors**. We can think of factors as the building blocks of a number. Consider the number 12, which can be written as 3 x 4, or 6 x 2, or 12 x 1. The factors of 12 are 1, 2, 3, 4, 6, and 12. Numbers always have themselves and '1' as factors since any number can be written as itself times 1.

One reason that it is useful to break a number into its factors is because it makes multiplication easier. If we multiply 12 x 36, we can use the fact that 12 = 3 x 4, 36 = 6 x 6, and 6 = 3 x 2 to see that:

- 12 x 36 = (3 x 4) x (6 x 6)
- 12 x 36 = 3 x 4 x 3 x 2 x 3 x 2

Do you see how we've just written a complicated multiplication problem as many small problems? This is one of the benefits of factoring.

**Prime numbers** are numbers that can be divided cleanly by only themselves and 1. The first few prime numbers are 2, 3, 5, and 7. Can you find some more? Prime numbers are the essential building blocks of all numbers.

If we try to break apart the multiplication problem 7 x 13 as we did above, we will not be able to. This is because 7 and 13 are both prime numbers. There is no way to simplify them any further. In fact, any number can always be written as a product of primes. Why not try some out for yourself?

Sieves or 'sifters' are typically used for separating solid materials from liquids. They usually have a metal mesh, and you may have seen one in a kitchen. The **sieve of Eratosthenes** works in much the same way - except instead of pouring liquid through it, you pour numbers. In order to apply the sieve, just follow these simple steps:

1. Begin with a large set of numbers that you would like to remove non-primes from, and put them in order.

2. Take the first non-prime that you find (such as 2), and remove all of the multiples of that number from the set ( 4, 6, 8...).

3. Repeat the second step with the next prime number.

Sounds pretty simple right? Let's try an example. Suppose we want to find all of the primes in the numbers from 1 to 15. We'll begin with {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.

1. Start by removing 1, since it is not considered a prime. Now remove all of the multiples of 2 because if something is a multiple of 2, it cannot be a prime. You'll have: {2, 3, 5, 7, 9, 11, 13, 15}.

2. Now remove all the multiples of 3, but leave 3 alone: {2, 3, 5, 7, 11, 13, 15}.

3. Now remove all of the multiples of 5. There is only one: 15. This leaves you with: {2, 3, 5, 6, 11, 13}.

Since there are no multiples of 6, 11, or 13 left, these are all of the primes that are less than 15. The sieve method can be used on much larger sets of numbers as well; it works perfectly no matter how many numbers you begin with.

Bonus Question: Does it matter if we put the numbers in order first? What would happen if we didn't?

In this lesson, you learned about using the ** sieve of Eratosthenes**, a method for finding **prime numbers**. We also discussed **factors** and how numbers can be broken down as the product of smaller numbers. Any number always has itself and '1' as factors.

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Math for Kids23 chapters | 325 lessons

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- What is an Abundant Number?
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- Finding the Sum of Consecutive Numbers 4:10
- What is a Cubed Number?
- What is a Deficient Number?
- What Are Figurate Numbers? - Definition & Examples
- What Are Opposite Numbers? - Definition & Examples 2:53
- What Are Rectangular Numbers? - Definition & Examples 3:05
- Like & Unlike Terms
- Is Zero an Integer?
- Is Zero a Natural Number? 2:37
- Sieve of Eratosthenes: Lesson for Kids
- Go to Types of Numbers for Elementary School

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