In the following exercises, find the prime factorization of each number using the factor tree method.
The Correct Answer and Explanation is:
Example: Find the prime factorization of 180 using the factor tree method.
We begin by breaking 180 into two factors:
180 = 18 × 10
Now factor each number further:
18 = 2 × 9
9 = 3 × 3
So, 18 = 2 × 3 × 3
10 = 2 × 5
Now list all the prime numbers:
180 = 2 × 3 × 3 × 2 × 5
Rewriting with similar terms together:
180 = 2 × 2 × 3 × 3 × 5
Final prime factorization:
180 = 2² × 3² × 5
Explanation:
The factor tree method is a way to break down a number into its smallest building blocks known as prime numbers. Prime numbers are whole numbers greater than 1 that can only be divided evenly by 1 and themselves. Examples include 2, 3, 5, 7, 11, and so on.
To use the factor tree method, start by picking any two factors of the number you are working with. Continue breaking down any non-prime numbers into smaller factors. Eventually, every branch of your tree will end in a prime number. These are the numbers that cannot be divided further except by 1 and themselves.
Once all numbers at the ends of your branches are prime, list them out. If a prime number appears more than once, you can write it using exponents for simplicity. This final product of prime numbers is called the prime factorization.
This method is helpful in many areas of math such as finding the greatest common divisor, least common multiple, simplifying fractions, and understanding number properties. The tree visually organizes the steps, making it easier to avoid mistakes. You can start the factor tree in different ways, but the final prime factorization will always be the same.
