### Lesson 4-4: Using Prime Factorization

Prime Numbers

A Prime Number is a whole number greater than one and can be divided evenly only by 1 or itself.

Let me explain through the following example …

*Example:*

**Composite Number**

When a number can

**not**be divided up evenly it is a

**Prime Number**

So **6** is Composite, but **7** is Prime.

**And that explains it … but I have some some more details to mention …**

The first few prime numbers are: 2, 3, 5, 7, 11, 13, and 17 …, and we have a prime number chart if you need more.

*Remember* that prime numbers can be only Whole Numbers (1, 2, 3, 4, 5 … etc), not fractions or negative numbers.

What About 1?

Years ago 1 was included as a Prime, but now **it is not**, so 1 is neither Prime nor Composite.

Factors

“Factors” are the numbers you multiply together to get another number: (**2 × 3 = 6**) here **2** and **3** are factors of **6**

Prime Factorization

“Prime Factorization” is a way to find which prime numbers multiply together to make a composite number.

An easy way to factor a number is by making a factor tree.

To make a tree, simply write the number you want to factor at the top of your paper.

From there, make branches of factors – numbers that multiply to give you the original number.

Next, take each of those numbers and break those down into more factors.

Continue until all the remaining numbers are prime numbers and cannot be factored anymore.

Have trouble understanding that explanation? We havean example below:

*Example:* What are the prime factors of 108 ?

**Please note**: You do not have to choose 6 and 18 as the first two “branches” of the tree. You can pick any two numbers that multiply to make 108. Other possibilities are like what is shown in the other trees 54 & 2, 36 & 3, and so on.

Now we can easily see the factors of 108 (the numbers inside the **green circles**) are 3, 3, 3 2, and 2.

All this means that if you take those numbers and multiply them together: (3)(3)(3)(2)(2) = 108.

We can also write this in a fancier way using exponents: 108 = 2^{2}•3^{3}

Factor Tree Game

Find the prime factors using factor tree.

Click user number to find prime factors of a specific number.

Click new number to randomly generate a number.

Drag the numbers, if they overlap.

Why prime factorization is important?

Cryptography is the study of secret codes. Prime Factorization is very important to people who try to make (or break) secret codes based on numbers. That is because factoring very large numbers is very hard, and can take computers a long time to do. If you want to know more, the subject is “encryption” or “cryptography”.

And here is another thing:

There is only one (unique!) set of prime factors for any number.

Example The prime factors of 330 are 2, 3, 5 and 11:

330 = 2 × 3 × 5 × 11

There is no other possible set of prime numbers that can be multiplied to make 330.

In fact this idea is so important, it is called the Fundamental Theorem of Arithmetic, which states that Any integer greater than 1 is either a **prime number**, or can be written as a **unique product of prime numbers** (ignoring the order).

What follows is a PowerPoint presentation that will help you understand the content of the lesson in a better and easier way, (*pps*).