Prime Factorization Using Factor Trees

Every composite number can be broken down into a unique set of basic building blocks called prime numbers. The process of finding these building blocks is called prime factorization.

One of the easiest and most visual ways to find the prime factorization of a number is by using a factor tree.

What are Prime and Composite Numbers?

Before we build a factor tree, let's quickly review:

• Prime Number: A number greater than $1$ that only has two factors: $1$ and itself. Examples include $2, 3, 5, 7, 11$, and $13$.
• Composite Number: A number that has more than two factors. Examples include $4, 6, 8, 9$, and $12$.

How to Build a Factor Tree

To find the prime factorization of a number using a factor tree, follow these simple steps:

1. Write the number at the top: Start with your target composite number.
2. Split it into two factors: Find any two numbers that multiply together to give you that top number. Draw two branches pointing down to these factors.
3. Check for primes: If either of the new numbers is a prime number, circle it. That branch is finished!
4. Keep branching: If a number is composite, split it again into two factors.
5. Stop when all branches end in a circle: Once every branch ends in a circled prime number, you are done.
6. Write the final equation: Multiply all the circled prime numbers together. We usually write them in order from smallest to largest and use exponents for repeated numbers.

Example 1: Find the prime factorization of 72

Let's break down the number $72$ using a factor tree.

1. Start with $72$.
2. What multiplies to $72$? Let's use $8$ and $9$. $72 = 8 \times 9$
3. Neither $8$ nor $9$ is prime, so we break them both down.
   • For $8$: $8 = 2 \times 4$. The number $2$ is prime, so we circle it. The number $4$ is composite.
   • For $9$: $9 = 3 \times 3$. The number $3$ is prime, so we circle both $3$s.
4. We still have that $4$. Let's break it down: $4 = 2 \times 2$. Circle both $2$s.

Now, collect all the circled prime numbers: $2, 2, 2, 3$, and $3$.

Write it as a product of primes: $72 = 2 \times 2 \times 2 \times 3 \times 3$

Using exponents, we can write this more neatly: $72 = 2^3 \times 3^2$

Example 2: Express 120 as a product of prime factors

Let's try a larger number, $120$.

1. Start with $120$.
2. Split it into two factors. Let's use $10$ and $12$. $120 = 10 \times 12$
3. Break down $10$: $10 = 2 \times 5$. Both $2$ and $5$ are prime. Circle them!
4. Break down $12$: $12 = 3 \times 4$. The number $3$ is prime (circle it), but $4$ is composite.
5. Break down $4$: $4 = 2 \times 2$. Both $2$s are prime. Circle them!

Collect all the circled primes: $2, 5, 3, 2, 2$.

Order them from smallest to largest and write the final product: $120 = 2 \times 2 \times 2 \times 3 \times 5$

With exponents, the prime factorization is: $120 = 2^3 \times 3 \times 5$

Tip: It doesn't matter which two factors you start with at the top of your tree. For $120$, you could have started with $2 \times 60$ or $12 \times 10$. As long as you keep going until only prime numbers are left, you will always get the exact same final answer!