Factor Trees: How to Find Prime Factorizations

What Is a Factor Tree?

A prime factor tree begins with a single composite number at the top. Two branches are drawn downward to a pair of factors that multiply together to equal the original number.

If either of these new factors is a prime number, it is circled to show that that branch has ended. If a factor is composite, it is branched again into two smaller factors.

The process stops only when every final branch ends in a circled prime number.

This method guarantees that all the foundational prime building blocks of a number are found without missing any values.

When to Use It

Factor trees are the most reliable tool for finding the prime factorization of large composite numbers.

Once a number is written as a product of its prime factors, it becomes much easier to calculate the greatest common factors and least common multiples between multiple numbers.

These relationships are essential when simplifying fractions or adding fractions with different denominators.

Step-by-Step Method

To draw a factor tree and find the prime factorization of a number, follow a consistent sequence.

1. Write the composite number at the top of the workspace.
2. Find any two factors that multiply together to make that number. Draw two branches downward to these factors.
3. Check each new factor. If a factor is a prime number, draw a circle around it. This branch is complete.
4. If a factor is composite, draw two new branches downward to a new pair of factors.
5. Repeat the branching process until every endpoint is a circled prime number.
6. Multiply all the circled prime numbers together to write the prime factorization.

Using divisibility rules can help you quickly find the first pair of branches. For example, if the number ends in $0$, you can immediately use $10$ as one of the factors.

Visual Worked Examples

The shape of a factor tree depends on the size of the number and the initial factors you choose. Larger numbers require deeper branches to reach all the prime numbers.

Example 1: Finding prime factors

Question: What is the prime factorization of $48$?

Method:

1. Write $48$ at the top of the tree.
2. Choose any two factors of $48$, such as $6$ and $8$.
3. Break down $6$ into $2$ and $3$. Since both are prime, circle them.
4. Break down $8$ into $2$ and $4$. Circle the $2$. Break $4$ into $2$ and $2$ and circle them.

Answer: The prime factors are $2$, $2$, $2$, $2$, and $3$. In index notation, $48 = 2^4 \times 3$.

Check: Multiply the primes together: $2 \times 2 = 4$, $4 \times 2 = 8$, $8 \times 2 = 16$, and $16 \times 3 = 48$.

Example 2: Repeated prime factors

Question: Use a factor tree to find the prime factorization of $100$.

Method:

1. Choose a factor pair for $100$, such as $10 \times 10$.
2. Write $100$ and branch it to $10$ and $10$.
3. Branch the left $10$ into $2$ and $5$. Both are prime, so circle them.
4. Branch the right $10$ into $2$ and $5$. Both are prime, so circle them.

Answer: Written from smallest to largest, $100 = 2 \times 2 \times 5 \times 5 = 2^2 \times 5^2$.

Check: Ensure all circled numbers are actually prime. $2$ and $5$ both have only two factors, so the branching is complete.

Example 3: Three-layer prime branches

Question: What is the prime factorization of $210$?

Method:

1. Begin with $210$. Since it ends in $0$, it is divisible by $10$.
2. Branch $210$ into $21$ and $10$.
3. Branch $21$ into $3$ and $7$. Both are prime. Circle them.
4. Branch $10$ into $2$ and $5$. Both are prime. Circle them.

Answer: Sorting the primes from smallest to largest, $210 = 2 \times 3 \times 5 \times 7$.

Check: Multiply sequentially to confirm the total: $2 \times 3 = 6$, $6 \times 5 = 30$, and $30 \times 7 = 210$.

How to Check the Answer

To verify your factor tree is correct, multiply all the circled prime numbers together.

The final product must exactly equal the number at the top of the tree. If the result is smaller, you may have forgotten to write down one of the primes. If the result is larger, you may have included a composite number in your final list or used an incorrect initial factor pair.

Common Mistakes

When building a factor tree, watch out for these frequent errors that lead to incorrect factorizations.

Using $1$ as a factor

Because $1$ is not a prime number, placing $1$ and the number itself on the branches creates an endless loop. A branch splitting $12$ into $1$ and $12$ requires you to split $12$ again, achieving no progress. Never use $1$ in a factor tree.

Adding instead of multiplying

The branches on a factor tree must represent multiplication, not addition. Splitting $24$ into $12$ and $12$ is incorrect because $12 \times 12 = 144$. The correct split requires finding two numbers that multiply to $24$, such as $3$ and $8$.

Stopping before all branches are prime

It is easy to circle a number like $9$ or $21$, mistaking it for a prime number. Always double-check your circled endpoints. If a circled number can be divided by $2$, $3$, $5$, or any other integer besides $1$, it is composite and the branch must be extended.