Factor Trees – A Visual Way to Find Prime Factors
A factor tree is a diagram that breaks a number down into its prime factors step by step. Each branch splits a number into two factors, and branching continues until every tip of the tree is a prime number.
How to Draw a Factor Tree
- Write the starting number at the top.
- Choose any pair of factors that multiply to give it (not 1 and the number itself).
- Write each factor as a branch below.
- If a factor is composite, split it again. If it is prime, circle it — that branch is complete.
- Collect all circled primes. That is the prime factorization.
Example 1 – Factor Tree for 48
48 splits to 6 × 8. 6 → 2 × 3 (both prime). 8 → 2 × 4 → 4 splits to 2 × 2. Primes: 2, 3, 2, 2, 2 → 48 = 2⁴ × 3
Example 2 – Factor Tree for 180
180 → 9 × 20 → 9 = 3 × 3; 20 → 4 × 5 → 4 = 2 × 2. Primes: 3, 3, 2, 2, 5 → 180 = 2² × 3² × 5
Different Trees, Same Answer
You can start by splitting in different ways and still get the same answer. That is the power of the Fundamental Theorem of Arithmetic.
Factor tree for 24 — two starting splits
Split A: 24 = 4 × 6 → 2×2 × 2×3 = 2³ × 3
Split B: 24 = 3 × 8 → 3 × 2×4 → 3 × 2 × 2×2 = 2³ × 3
Same result: 2³ × 3
Tips for Accuracy
- Always try to spot an even factor first — dividing by 2 is easiest.
- Circle primes as you go so you don't lose track.
- Count the prime branches at the end to double-check by multiplying back up.
Key Takeaways
- Any starting split works — the final prime factorization is always the same.
- A branch ends only when the tip is a prime number.
- Multiply all the circled primes to verify you get the original number back.
Practice Questions
- Draw a factor tree for 60 and write the prime factorization.
- Draw a factor tree for 126.
- Two students draw different trees for 90 but get the same factorization. What is it?
- Draw a factor tree for 256.
- Use a factor tree to find GCF(36, 84).