Direct answer: A factor tree is a diagram that breaks a number into its prime factors: split it into two factors, then split those factors again, branch by branch, until every branch ends in a prime number. The primes at the ends — the leaves — multiplied together give the prime factorization. For 48, the leaves give 48 = 2⁴ × 3.
If you have ever been told to “draw a factor tree” and pictured an actual tree, you are closer than you think. The diagram really does branch like a tree: the number sits at the top, two branches split it into factors, and each branch keeps splitting until it reaches a prime number. It is the most visual way to find prime factorization, and the method teachers reach for first in the classroom.
In this guide, you will learn how to draw a factor tree in five easy steps, see two fully worked examples, and learn the two facts that make the method click: any starting pair gives the same answer, and the one mistake that ruins most trees. You can check any answer with Calcrio’s free tool, which gives the prime factorization of any number instantly — plus its full factor list, factor pairs, and the GCF and LCM of two numbers.
What Is a Factor Tree?
A factor tree is a branching diagram used to find the prime factorization of a number. It has three parts:
- The trunk: the number you start with, written at the top.
- The branches: each split of a number into two factors, drawn as two lines going down.
- The leaves: the prime numbers at the ends of the branches. A prime number cannot be split further, so it is a leaf — done, circled, finished.
When every branch ends in a leaf, you are done. Multiply all the leaves together and you get the prime factorization.
Why does it work? Because of a simple fact: every whole number greater than 1 has exactly one prime factorization. No matter which factor pair you start with, and no matter how you split the branches, the leaves always come out the same. That is the guarantee that makes the factor tree trustworthy — there is one right answer, and any correct tree finds it.
How to Draw a Factor Tree in 5 Easy Steps
Here is the method, exactly as it works on paper:
- Write the number at the top of your page. This is the trunk of your tree.
- Split it into any two factors. Draw two branches down to the two factors. Any pair works — pick the easiest one you can see.
- Circle any prime numbers. A prime factor is a leaf: circle it and leave it alone. It does not split further.
- Split any composite number again. If a factor is not prime, split it into two factors of its own and draw two more branches.
- Collect the leaves. When every branch ends in a circled prime, write the primes in a row and multiply them together. That is the prime factorization.
That is all there is to it. Now let us draw two real trees.
Worked Example: A Factor Tree for 48
Start with 48 at the top. Split it into 8 × 6 — both easy to see. Now 8 is not prime, so split it: 8 = 2 × 4. And 6 is not prime either: 6 = 2 × 3. The 2 and 3 are primes, so circle them. The 4 is still composite: 4 = 2 × 2, and both 2s are prime, so circle them too.
Here is the tree as it looks on paper:
48
/ \
8 6
/ \ / \
2 4 2 3
/ \
2 2
Read the leaves from left to right: 2, 2, 2, 2, and 3.
The prime factorization of 48 is 2 × 2 × 2 × 2 × 3 = 2⁴ × 3.
Check it: 2 × 2 × 2 × 2 = 16, and 16 × 3 = 48. Correct. You can check any tree with Calcrio’s free factor tree tool.
Now here is the fact that surprises most students. Try a completely different start: split 48 into 12 × 4 instead of 8 × 6. Then 12 = 3 × 4, 4 = 2 × 2, and the other 4 = 2 × 2. The leaves are 3, 2, 2, 2, and 2 — the same five primes, in a different order. Any starting pair gives the same answer. That is the one-prime-factorization guarantee, and it means you can never “pick the wrong pair” — you just pick the pair you see first.
Worked Example: A Factor Tree for 120
120 is a bigger number, so the tree has more branches — and that is where the method earns its keep.
Start with 120 = 12 × 10. Split 12 into 2 × 6 (2 is prime, circle it; 6 splits into 2 × 3, both prime). Split 10 into 2 × 5 (both prime).
120
/ \
12 10
/ \ / \
2 6 2 5
/ \
2 3
The leaves are 2, 2, 3, 2, and 5.
The prime factorization of 120 is 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5.
Check it: 2 × 2 × 2 = 8, 8 × 3 = 24, and 24 × 5 = 120. Correct.
Notice how the tree handles a big number without stress: at every step you only split one number into two factors, and the biggest number you deal with is the one you started with. It turns one hard problem into a chain of easy ones.
How to Write Prime Factorization from a Factor Tree
Once the tree is drawn, the last step is writing the answer properly. Collect the leaves, count how many times each prime appears, and write the result using exponents:
- For 48: the leaves are 2, 2, 2, 2, 3 → 48 = 2⁴ × 3.
- For 120: the leaves are 2, 2, 2, 3, 5 → 120 = 2³ × 3 × 5.
- For 60: the leaves are 2, 2, 3, 5 → 60 = 2² × 3 × 5.
Exponents are not decoration — they are how the answer is used. The exponents tell you how many factors the number has (add 1 to each exponent and multiply), and they are how you find the GCF and LCM of two numbers. For example, 48 = 2⁴ × 3 has (4 + 1) × (1 + 1) = 10 factors. Our guide to prime factorization methods explains the division method as well, so you can compare the two and pick the faster one per problem.
The #1 Mistake: Adding Instead of Multiplying
The most common factor-tree mistake is also the easiest to make: adding the leaves instead of multiplying them. For 48, the leaves are 2, 2, 2, 2, and 3. Their sum is 11. Their product is 48. Only the product is the answer.
The fix: always verify by multiplying your leaves back together. If the product is not the number you started with, something is wrong — usually a missed branch or a composite number you forgot to split.
Two more mistakes to watch for:
- Stopping too early. If a branch ends in a composite number like 4, 6, or 9, the tree is not finished. Only primes are leaves. Circle only primes.
- Forgetting a branch. In a wide tree like 120, it is easy to split one side and forget the other. Work left to right, one branch at a time, and circle each prime as you go.
Factor Tree vs Listing Factors: What Is the Difference?
Students sometimes confuse factor trees with finding all the factors of a number. They are different jobs:
- Finding factors lists every number that divides into the target: the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48 — ten factors in total.
- A factor tree finds only the prime building blocks: 48 = 2⁴ × 3.
The tree is the faster route to prime factorization, and the factor list is what you need for factor pairs and for GCF problems. Both start from the same skill — splitting a number into factors — and the free tool at Calcrio gives you both answers at once: the complete list of factors of a number, its factor pairs, and its prime factorization.
Why Teachers Love Factor Trees
Factor trees are a classroom favorite for three reasons: they are visual — a student who struggles with divisions can see the branches and follow them; they are forgiving — any starting pair works, so there is no wrong first move; and they lead naturally into the next topics, since the leaves of two trees are exactly what you compare to find the GCF and LCM, as our GCF vs LCM guide shows with 12 and 18.
That is also why factor trees cross over so well to Pinterest and classroom printables: the diagram itself is the teaching.
Frequently Asked Questions
What is a factor tree?
A factor tree is a branching diagram that breaks a number into its prime factors. You split the number into two factors, then keep splitting each factor until every branch ends in a prime number. The primes at the ends — the leaves — multiplied together give the prime factorization.
How do you make a factor tree?
Write the number at the top, split it into any two factors, and draw two branches. Circle any prime factors, and split any composite factors again. Repeat until every branch ends in a prime, then multiply the circled primes together for the prime factorization.
What are the leaves of a factor tree?
The leaves are the prime numbers at the ends of the branches. A prime cannot be split further, so it is a finished leaf. For 48, the leaves are 2, 2, 2, 2, and 3, giving 48 = 2⁴ × 3.
Does the factor tree always give the same answer?
Yes. Every whole number greater than 1 has exactly one prime factorization, so any correct factor tree ends in the same primes. Starting with 8 × 6 or 12 × 4 for 48 both give the leaves 2, 2, 2, 2, and 3 — just in a different order.
How do you write prime factorization from a factor tree?
Collect the circled prime leaves, count how many times each prime appears, and write the result with exponents. For 120, the leaves 2, 2, 2, 3, 5 become 120 = 2³ × 3 × 5.
What is the most common factor tree mistake?
Adding the leaves instead of multiplying them. For 48, the leaves 2, 2, 2, 2, and 3 add up to 11 but multiply to 48. Always verify by multiplying your leaves back together — the product must equal the number you started with.