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Prime Factorization



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In this section, we learn how to find the prime factorization of a number. We start out by learning about the difference between prime and composite numbers. A prime number is a whole number that is larger than 1, and only divisible by itself or 1. We will find that 2 is the only even prime number. Every even number larger than 2 will be divisible by 2 and therefore not fit the definition of a prime number. A composite number is a whole number that is greater than 1 and divisible by some number other than itself and 1. The numbers 0 and 1 do not fall into either category. They are not considered to be prime numbers or composite numbers.
Example 1: Determine if 5, 20, and 32 are prime, composite, or neither
5 - prime, it is only divisible by itself 5 or 1
20 - composite, the number is even and larger than 2
32 - composite, the number is even and larger than 2
Next, we will think about how to break a number down into the product of its prime factors. Breaking a number down into the product of prime factors is known as finding its prime factorization. This process is very simple and in most cases, one can use a visual aid such as a factor tree or division ladder.
Prime Factorization of a Number:
  • Start with any two factors of the number
  • For each factor we check to see if the number is prime, circle any prime factor(s) and stop working on that/those numbers
  • For factors that are composite, we repeat the process and find any two factors
  • This process continues until all factors are prime
  • The circled numbers represent the prime factors of the number
Example 2: Find the Prime Factorization of 120
Start with any 2 factors of the number: 120 = 60 x 2
Check to see if either factor is prime: 60 is not prime, 2 is prime
Circle 2 and continue working on 60
120 = 60 x 2
Repeat the process on 60, find any 2 factors: 60 = 30 x 2
Check to see if either factor is prime: 30 is not prime, 2 is prime
Circle 2 and continue working on 30
120 = 30 x 2 x 2
Repeat the process on 30, find any 2 factors: 30 = 15 x 2
Check to see if either factor is prime: 15 is not prime, 2 is prime
Circle 2 and continue working on 15
120 = 15 x 2 x 2 x 2
Repeat the process on 15, find any 2 factors: 15 = 5 x 3
Check to see if either factor is prime: 5 is prime, 3 is prime
Circle 5 and 3, we have found our prime factorization:
120 = 5 x 3 x 2 x 2 x 2
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