Concept 2 of 4
The Roots of Numbers: Prime and Composite
Today's big idea
A prime refuses to split, and every other number is built by multiplying primes.
12 splits into 2 x 6, then the 6 into 2 x 3, so 12 = 2 x 2 x 3.
How it works
👀 What's going on?
Some numbers refuse to be split up. 7 can only be built as 1 x 7, and the same goes for 11 and 13. Others come apart easily, and 12 comes apart in three different ways.
💡 Here's the trick
Look at the tree above. 12 splits into 2 x 6, and then the 6 splits again into 2 x 3. Now nothing splits further, because 2 and 3 are prime. Every composite number breaks down into primes like this, and the primes are simply where the splitting has to stop. Those end numbers are the leaves of your tree.
🪄 How to do it
- 1.Try the small primes in turn, 2 then 3 then 5 then 7, the way the picture above starts with 2.
- 2.If one of them divides evenly the number is composite, so split it and carry on down.
- 3.If none of them divide it, the number is prime and the tree ends there.
✍️ Try it
Is 21 prime or composite? And what about 23? Try the small primes on each of them.
For 21, the 2 does not go but the 3 does, giving 21 = 3 x 7, so 21 is composite. For 23, none of 2, 3 or 5 divide it evenly, so 23 is prime. Notice that finding a single divisor was enough to settle 21 straight away.
⚠️ Watch out
You might be thinking 1 must be prime, since nothing looks less splittable. It is not. A prime has exactly two different factors, and 1 only has itself. The smallest prime is 2, and it is the only even one there is.
🌟 Why this matters
Primes are the building blocks every other number is made from, rather as letters build words. They are also what keeps online payments safe, because very large primes are extremely hard to pull apart.
Exam tip: Learn the primes below 50 by heart, because factor, HCF and LCM questions all lean on them.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Is 15 a prime number?”
- 1
Try dividing by small primes
Start with 2 — 15 ÷ 2 doesn't work (15 is odd).
- 2
Try 3
15 ÷ 3 = 5
- 3
It divided! So 15 isn't prime
15 = 3 × 5 — found a factor besides 1 and 15.
- 4
Verdict ✓
15 is COMPOSITE (3 × 5)
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Prime numbers split no further (2, 3, 5, 7…). Composite numbers break into smaller factors. Keep splitting until every leaf is prime!
PRIME vs COMPOSITE
Prime: only 1 × itself (7 = 1×7 only)
Composite: has other factors (12 = 2×6, 3×4…)
Tap the two COMPOSITE numbers:
7 and 5 are prime — leave them alone.
Quick check
✅ VERIFICATION
Which of these numbers is prime?
⚡ Finish the activity above first to answer.
Keep going!
Up next: HCF & LCM
HCF is the biggest factor they share; LCM the smallest multiple.