Prime Factorization - Factoring Whole Numbers Lesson

What is a prime number?

In our last lesson, we learned about the divisibility rules. These rules allow us to quickly determine if a number is divisible by another. This means the division will have no remainder. When we think about the whole numbers, we can separate these numbers into three groups:

• Prime Numbers - a whole number larger than 1 that is only divisible by itself and 1.
• Composite Numbers - a whole number larger than 1 that is divisible by a whole number other than itself and 1.
• The whole numbers 0 and 1 are not prime or composite:
  • They are not in either category

The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The set of prime numbers is infinite and continues forever. The smallest prime number is 2, and it is the only even prime number. This is due to the fact that any even number larger than 2 is divisible by 2. This means the number is divisible by some number other than itself and 1. To check and see if a number is prime or composite, we can run through the divisibility rules for the number. We only have to check up to the square root of the number. We haven't yet discussed square roots, for now, we can access this from most calculators. Let's think about a few examples.
Example 1: Determine if each number is prime, composite, or neither.
21

• Using the square root key on a calculator gives us an approximate value of 4.58; we will check the divisibility of 21 by 2, 3, and 4.
• First, we notice the number is not even. This means it will not be divisible by 2, or any other even number.
• Next, we think about divisibility by 3. For divisibility by 3, the sum of the number's digits must be divisible by 3. 2 + 1 = 3 and 3 ÷ 3 = 1. This means the number 21 is divisible by 3. 21 ÷ 3 = 7.
• 21 is a composite number since it is divisible by some number other than itself (21) and 1.

17

• Using the square root key on a calculator gives us an approximate value of 4.12; we will check the divisibility of 17 by 2, 3, and 4.
• First, we notice the number is not even. This means it will not be divisible by 2 or any even number.
• Next, we think about divisibility by 3. For divisibility by 3, the sum of the number's digits must be divisible by 3. 1 + 7 = 8 and 8 ÷ 3 = 2 R2. This means the number 17 is not divisible by 3.
• 17 is a prime number since it is not divisible by some number other than itself (17) and 1.

143

• Using the square root key on a calculator gives us an approximate value of 11.96; we will check the divisibility of 143 by the numbers 2 - 11.
• First, we notice the number is not even. This means it will not be divisible by 2 or any even number.
• Next, we think about divisibility by 3. For divisibility by 3, the sum of the number's digits must be divisible by 3. 1 + 4 + 3 = 8 and 8 ÷ 3 = 2 R2. This means the number 143 is not divisible by 3. If a number is not divisible by 3, it can't be divisible by 9. We can cross 9 off of our list as well.
• Next, we think about divisibility by 5. For divisibility by 5, the final digit needs to be a 0 or a 5. This is not the case here since our final digit is a 3. We can say that 143 is not divisible by 5. Since the number is not divisible by 5, it can't be divisible by 10. We can cross 10 off of our list as well.
• Next, we think about divisibility by 7. For divisibility by 7, we cut off the final digit of the number and double it. We then subtract this from the shortened number. 14 - 6 = 8. Since 8 is not divisible by 7 and not a 0, 143 is not divisible by 7.
• Next, we think about divisibility by 11. For divisibility by 11, form the sum of the digits in the odd places and subtract the sum of the digits in the even places. (1 + 3) - 4 = 0. Since this result is 0, we can say that 143 is divisible by 11. 143 ÷ 11 = 13.
• 143 is a composite number since it is divisible by some number other than itself (143) and 1.

How to Factor a Whole Number into the Product of Prime Numbers

When we start working with fractions, we will need to know how to factor a whole number in order to simplify. Once we have a full understanding of the definition of a prime number, we are ready to move into finding the prime factorization of a number. To find the prime factorization of a number, we are simply writing the whole number as the product of prime factors. There are many ways to achieve this result, but we will focus on building a factor tree.

How to Build a Factor Tree - Factoring Whole Numbers

• Write the number to be factored and extend two lines below. These lines are known as "branches".
• List any two factors of the number under the branches (excluding the number and 1).
• Look at each factor and determine if the number is prime. We circle any number that is prime.
• For any number that is not prime, we repeat the process.
• The prime factorization can be found as the product of all the circled numbers

Let's walk through an example step by step. Suppose we wanted to find the prime factorization of the number 120:

• Write the number to be factored and extend two lines below. These lines are known as "branches".

• List any two factors of the number under the branches (excluding the number and 1).
• We will select 30 and 4 as our factors. Remember, we can use any two factors.

• Look at each factor and determine if the number is prime. We circle any number that is prime.
• Neither number is prime.
• For any number that is not prime, we repeat the process.
• For the number 30, we will select 5 and 6 as our factors. For the number 4, we only have the choice of 2 and 2.

• We now have found 2 prime factors: 5 and 2. We will circle these numbers and stop on those branches.

• For the number 6, we only have the choice of 3 and 2. Both of these numbers are prime and will be circled.

• The prime factorization can be found as the product of all the circled numbers:
  • 3 x 2 x 5 x 2 x 2 or 2³ x 3 x 5

120 = 2³ x 3 x 5
Let's try a few examples.
Example 2: Find the prime factorization of 1050. 1050 = 2 x 3 x 5² x 7
Example 3: Find the prime factorization of 1320. 1320 = 2³ x 3 x 5 x 11

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