Introduction

In this lesson, you will learn the basic structure of prime factorization. A prime number can only be divided by one and itself.

These videos illustrate the lesson material below. Watching the videos is optional.

• Prime Numbers (05:52 mins) | Transcript
• Prime Factorization (06:44 mins) | Transcript

Identifying Prime Numbers

Learning how to find the prime factorization of a number is important when you start learning about fractions. It is especially important when reducing fractions. Here are some vocabulary words that will help you with this lesson:

• Factor: The numbers in a multiplication operation. (For example: In $3\times 4=12$, 3 and 4 are the factors of 12.)
• Product: The solution to multiplication.  (For example: In $3\times 4=12$, 12 is the product.)
• Prime Number: Any number where the only factors are 1 and itself (for example: 11 is a prime number. There is no other number other than 1 and 11 you can divide it by that will give you a whole number).

Figure 1

Using the multiplication chart above, consider the following numbers and learn how to determine whether a number is prime:

1. One is not considered prime; it is just the number one. There are reasons for that which aren't necessary to learn now. Just know that number one is not considered prime.
2. Two is $1\times 2$ and it’s also $2\times 1$, but those are the only instances where you see the number two on the multiplication chart. So, two is only the product of one and itself, which makes it a prime number.
   
   • Note: Number two is the only even number that is prime; all others are odd numbers. However, not every odd number is prime.
3. Four is the multiple of $1\times 4$, $4\times 1$, and $2\times 2$. Four can’t be a prime number because it has other factors—1, 2, 4. Remember, factors are numbers that, when multiplied together, create a product.
4. Seven is the product of $1\times 7$ and $7\times 1$. But again, those are the only two cases where three appears on the chart, which indicates that three is prime.

What are some other prime numbers on the chart? Figure 2 highlights the prime numbers in yellow up to number 139.

Figure 2

Prime Factorization

The smallest numbers you can break a number down into are prime numbers because a prime number cannot be broken down any further.

Example 1
14 isn’t prime because it is the same as $2\times 7$.

Figure 3

The factorization is $2\times 7$. To factor a number, break it down by listing the factors underneath, creating a tree-like shape. Then evaluate the factors. Because the factors 2 and 7 are both prime, this is as far as you can factor 14. Therefore, 2 and 7 are the prime factors of 14.

Example 2
There are lots of ways you can get 40, such as $8\times 5=40$ and $4\times 10=40$. This means that 40 can be factored multiple ways. First, try using the factors 4 and 10.

Figure 4

4 is not prime, so you can break it down even further. $2\times 2=4$, and 2 is prime, so that’s as far as you can go with the 4. The other factor, 10, is not prime either. 10 can be broken down into $2\times 5=10$, and 2 and 5 are both prime.

The prime factorization of 40 is $2\times 2\times 2\times 5$, or you could even put this in exponential notation as $2^3\times 5$. Both of these formats convey the correct answer.

Now try using the factors of 8 and 5 because 40 can also be broken down into $8\times 5$. The figure below shows this factorization.

Figure 5

Factoring it this way still got $2\times 2\times 2\times 5$ as the prime factors of 40.

Example 3
What is the prime factorization of 19? Look through the chart and see what multiples make up 19 before looking at Figure 6 below.

Figure 6

Because 19 doesn’t have any multiples other than 1 and 19, that is where factorization stops. 19 is a prime number.

Example 4
210 has a factor of 2 because it is an even number. $2\times 105$. 105 has a prime factor of 5 because it ends with 5: $5\times 21$. 21 has two prime factors of 3 and 7. So the prime factorization of 210 is $2\times 5\times 3\times 7$.

Figure 7

Once you know what prime numbers are, you learn that each number is made up of smaller prime numbers. Breaking a number down into the prime numbers that make it is called prime factorization. Every number has a prime factorization. For prime numbers, their only factors are themselves and 1.

Things to Remember

• Prime numbers are numbers that can only be divided by one and itself.
• Prime numbers are made up of odd numbers, except the number two. But remember that not every odd number is prime.
• Factors are numbers that create a product.
• Factoring out the number means you go all the way to the prime numbers. This can be done with any number you are given, even a prime number, although you won’t get very far.

Practice Problems

1. 21 (
   Solution

   x

   Solution: $21=3\times 7$
   Details:
   
   $3\times 7=21$; and 3 and 7 are prime factors. The prime factorization of the number 21 is 3 and 7.
   )
2. 13 (
   Solution

   x

   Solution: $13=1\times 13$
   Details:
   
   The definition of a prime number is any number where the only factors are 1 and itself.
   The only factors of 13 are 1 and itself, 13. The prime factorization of 13 is just 13.
   )
3. 30 (
   Video Solution

   x

   Solution: $30=2\times 3\times 5$
   Details:
   

   
   
   | Transcript)
4. 12 (
   Solution

   x

   Solution: $12=2\times 2\times 3$
   Details:
   First, find the factors of 12. There are a few ways to factor 12. It can be factored as either $4\times 3$ or $6\times 2$
   
   
   The numbers 3 and 2 are both prime, but 4 and 6 are not.
   Find the factors of 4 or 6. Notice how the prime factorization of 12 is $2\times 2\times 3$ whether the starting factors were $4\times 3$ or $6\times 2$. The prime factorization is the same.
   
   
   )
5. 54 (
   Solution

   x

   Solution: $54=2\times 3\times 3\times 3$
   Details: 54 is an even number so you know it is divisible by 2.
   $2\times 27=54$
   
   The number 2 is prime, but 27 is not, so you must find the factors of 27.
   27 is divisible by 3:
   $3\times 9=27$
   
   The number 3 is a prime number, but 9 is not prime, so you must find the factors of 9.
   9 is also divisible by 3:
   $3\times 3=9$
   
   The number 3 is a prime number so there are actually two more 3’s in this prime factorization.
   Now look back and find all the prime numbers in the factorization of 54.
   
   $54=2\times 3\times 3\times 3$
   )
6. 250 (
   Video Solution

   x

   Solution: $250=2\times 5\times 5\times 5$
   Details:
   

   
   
   | Transcript)