Prime Factorization is used to find the Least Common Multiple and the Greatest Common Factor. This lesson will address a few definitions, give how-to instructions, detailed examples, and share a website for online practice.

Prime Numbers
Prime numbers have only two factors. Those factors are one and itself. For example, 17 has the factors 1 and 17 only. Thus, 17 is a prime number. Likewise, 2, 3, 5, 7, 11, 13, 17, 19 are prime numbers too. The number 39 has factors 1, 3, 13, and 39. Thus, 39 is not a prime number.

Composite numbers
Numbers with other factors besides themselves and one are called composite numbers.
Therefore, 39 is a composite number. By the way, one is not a prime or composite number.

Prime Factorization ---------------------Usefulness: to find the Greatest Common Factor or Least Common Multiple

When a composite number is factored using only prime numbers such as 2 x 3 x 5 = 30, it is referred to as prime factorization.

How to Find the Prime Factorization of a Number

1) Start with the smallest prime number, 2, and ask yourself if the prime number can divide into the given number without a remainder. In other words, is it divisible by 2?

2) If no, then the prime number is not a factor. Try the next prime number.

3) If yes, then include that prime number in the prime factorization equation.

4) If the given number was divisible by the prime number in step one, was the answer a composite or prime number? If composite, use this number and repeat steps 1 – 3 starting with the prime number 2 again.

If the answer is a prime number, divide the number by itself to get one and you are finished; include all prime numbers in the factorization.

5) Check – compute the multiplication sentence and the answer should equal the number just factored.

Let’s find the prime factorization of 30

1) Start with the smallest prime number 2. Ask yourself if the prime number can divide into 30 without a remainder. 30 / 2 = 15 remainder 0

3) Yes it can. Then include 2 in the prime factorization equation.

4) In step 1,was the answer a composite or prime number? 15 is a composite number. So, repeat the process with 15 starting with 2 again.

15 / 2 = 7 remainder 1; 15 is not divisible by 2; so, 2 won’t be used again

Next, try 3; 15 / 3 = 5; 15 is divisible by 3; 3 becomes part of the factorization.

The answer 5 is prime; so, divide 5 by itself ----- 5 / 5 =1

You are finished; include all prime numbers in the factorization.
Summary:
30/ 2 = 15
15/ 3 = 5
5 / 5 = 1
The prime factorization of 30 = 2 x 3 x 5.

The bold numbers are the prime factors of 30.
Check – compute the multiplication sentence and the answer should equal the number just factored, 30.


Example 2: Find the prime factorization of 45

45 / 3 = 15
15 / 3 = 5
5 / 5 = 1
Prime factorization of 45 = 3 x 3 x 5


Example 3: Find the prime factorization of 88

82 / 2 = 44
44 / 2 = 22
22 / 2 = 11
11/ 11 = 1
Prime factorization of 88 = 2 x 2 x 2 x 11.

For online practice: I highly recommend the website in the related links section. It uses the factor tree method which is similar to the above method.