Concept 3 of 6
Prime Factorisation: The Factor Tree
Today's big idea
A factor tree splits a number until every leaf is prime and splits no further.
Splitting order does not matter, because the set of primes comes out the same.
How it works
👀 What's going on?
Every number above 1 is built out of primes, and there is only one recipe that works. A factor tree is how you dig that recipe out.
💡 Here's the trick
Look at the tree above. A prime has exactly two factors, 1 and itself, so it cannot be split any further. The primes below 20 are 2, 3, 5, 7, 11, 13, 17 and 19, and 1 is not one of them.
🪄 How to do it
- 1.Split the number into any two factors, as the tree above does.
- 2.Keep splitting every branch that is still composite.
- 3.Collect the prime leaves and write them as powers.
✍️ Try it
Build a factor tree for 90. Then: write 100 as a product of primes.
Splitting 90 into 9 and 10 leaves 3, 3, 2 and 5, so 90 = 2 × 3² × 5. Splitting 100 into 10 and 10 leaves 2, 5, 2 and 5, so 100 = 2² × 5².
⚠️ Watch out
You might be thinking a different first split gives a different answer. It cannot. Start 90 off as 2 and 45 instead, and the leaves are still 2, 3, 3 and 5.
🌟 Why this matters
Online payment codes are built from primes, which is what makes them so hard to crack. Closer to home, prime factors are the quickest way to simplify a fraction.
Exam tip: Write the primes smallest first, so none of them slips out of your final answer.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Write 60 as a product of primes (factor tree).”
- 1
Split into any two factors
60 = 6 × 10. 🌳
- 2
Keep splitting until prime
6 = 2 × 3 (both prime). 10 = 2 × 5 (both prime).
- 3
Collect the primes
2, 2, 3, 5 → write as powers: 2² × 3 × 5
- 4
Prime factorisation ✓
60 = 2² × 3 × 5
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Keep splitting each branch until every leaf is a prime number.
FACTOR TREE · ROUND 1 OF 3
36
Quick check
✅ VERIFICATION
What is the prime factorisation of 45?
⚡ Finish the activity above first to answer.
Keep going!
Up next: HCF
hcf of 12 and 18 = 6. Use it for equal-share problems.