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Factor Tree — Definition, Examples & How to Find

Factor Tree

A structure used to find the prime factorization of a positive integer.

 

Factor tree for 60: 60 splits into 3 and 20, 20 splits into 4 and 5, 4 splits into 2 and 2. Result: 60 = 2² · 3 · 5

 

 

See also

Factor

Worked Example

Problem: Use a factor tree to find the prime factorization of 60.
Step 1: Start with 60 at the top. Choose any two factors of 60 (other than 1 and 60 itself). For instance, 60 = 6 × 10.
60=6×1060 = 6 \times 10
Step 2: 6 is not prime, so break it into two factors: 6 = 2 × 3. Both 2 and 3 are prime, so circle them — these branches are done.
6=2×36 = 2 \times 3
Step 3: 10 is not prime, so break it into two factors: 10 = 2 × 5. Both 2 and 5 are prime, so circle them.
10=2×510 = 2 \times 5
Step 4: Collect all the circled primes at the ends of the branches: 2, 3, 2, and 5.
60=2×2×3×5=22×3×560 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5
Answer: The prime factorization of 60 is 22×3×52^2 \times 3 \times 5.

Another Example

Problem: Use a factor tree to find the prime factorization of 72.
Step 1: Start with 72. Split it into any factor pair: 72 = 8 × 9.
72=8×972 = 8 \times 9
Step 2: 8 is not prime. Break it down: 8 = 2 × 4. Then 4 is not prime either: 4 = 2 × 2. Now all three factors (2, 2, 2) are prime.
8=2×2×28 = 2 \times 2 \times 2
Step 3: 9 is not prime. Break it down: 9 = 3 × 3. Both are prime.
9=3×39 = 3 \times 3
Step 4: Collect all primes from the branch ends.
72=2×2×2×3×3=23×3272 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2
Answer: The prime factorization of 72 is 23×322^3 \times 3^2.

Frequently Asked Questions

Does it matter which pair of factors you start with in a factor tree?
No. You can split the number into any valid factor pair, and you will always arrive at the same set of prime factors at the end. This is guaranteed by the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 has exactly one prime factorization (up to the order of the factors). Your tree may look different, but the final answer will be the same.
When do you stop branching in a factor tree?
You stop branching whenever you reach a prime number. A prime has no factors other than 1 and itself, so it cannot be split further. Once every end of every branch is a prime, the tree is complete.

Factor Tree vs. Repeated Division (Ladder Method)

A factor tree splits a number into two factors at each step and arranges them as branches in a tree diagram. The repeated division method (sometimes called the ladder or upside-down division method) divides the number by the smallest prime factor repeatedly, stacking results vertically. Both methods produce the same prime factorization. Factor trees offer more flexibility because you can choose any factor pair at each step, while repeated division is more systematic since you always divide by the smallest available prime.

Why It Matters

Factor trees give you a visual, organized way to break any composite number down to its prime building blocks. Once you know a number's prime factorization, you can easily find the greatest common factor (GCF) or least common multiple (LCM) of two or more numbers. Prime factorization also appears in simplifying fractions, working with radicals, and understanding divisibility.

Common Mistakes

Mistake: Stopping a branch at a composite number (e.g., treating 4 or 9 as if they were prime).
Correction: Always check whether each number at the end of a branch is prime. If it can be divided by any number other than 1 and itself, keep splitting. For example, 4 = 2 × 2 and 9 = 3 × 3.
Mistake: Forgetting to include a repeated prime factor when writing the final answer.
Correction: Carefully collect every prime at every branch end, including duplicates. For instance, the factor tree for 60 produces two 2s, one 3, and one 5 — so the factorization is 22×3×52^2 \times 3 \times 5, not 2×3×52 \times 3 \times 5.

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