How to find prime factors with a factor tree
Every whole number bigger than 1 is either a prime or can be made by multiplying primes. 12 is 2 × 2 × 3. Those primes are its prime factors, and finding them is called prime factorisation. A factor tree is one way to do it.
Split the number into two factors. Keep splitting until every branch ends in a prime. The primes at the ends of the branches are the prime factors.
How to do it
- Write the number at the top.
- Think of two numbers that multiply to make it. Write them underneath, on two branches. Any pair will do, except 1 and the number itself.
- If a branch is a prime, circle it. That branch is finished.
- If a branch is not prime, split it into two factors again.
- When every branch ends in a circle, multiply the circled primes together. Write them smallest first.
Worked examples
What are the prime factors of 60?
Every branch ends in a prime, so 60 = 2 × 2 × 3 × 5.
What are the prime factors of 36? Start with 4 × 9.
36 = 2 × 2 × 3 × 3.
What if we start 36 with 6 × 6 instead?
A different tree, but the same primes. 36 = 2 × 2 × 3 × 3 again.
Stuck for a way to split it?
Split off the smallest prime you can. Is the number even? Split off a 2. Do its digits add up to a multiple of 3? Split off a 3. Does it end in 0 or 5? Split off a 5. The divisibility rules tell you quickly.
The factor ladder works this way every time, and writes it as a column instead of a tree.
Why any tree gives the same answer
Every number has only one set of prime factors. The tree can grow in different shapes, but the primes at the ends are always the same. That is why you can start with any split you like.
To check your answer, multiply the primes back together: 2 × 2 × 3 × 5 = 60.
Where mistakes happen
- Stopping at a number that is not prime. 6 is not finished, it still splits into 2 × 3.
- Splitting off a 1. 60 = 1 × 60 gets you nowhere, and 1 is not prime.
- Losing a prime when writing the answer. Count the circles, and multiply back to check.
Questions people ask
Do I have to start with a particular pair?
No. Any two factors work. Small primes like 2, 3 and 5 are usually the easiest.
Why write the primes smallest first?
It is not wrong in another order, because 5 × 3 × 2 × 2 is still 60. Smallest first just makes answers easy to compare.
How do I write 2 × 2 × 3 × 5 more shortly?
Use a power: 2² × 3 × 5. The small 2 means "two 2s multiplied". This is called index form.
What if the number is prime?
Then it cannot be split. Its tree is just the number, circled.