Concept 4 of 6
HCF: The Highest Common Factor
Today's big idea
The highest common factor is the biggest number dividing both of them exactly.
Common means it sits in both lists, and highest picks the largest of those.
How it works
👀 What's going on?
Two piles of different sizes, both to be split into equal groups. The biggest group size that works for both piles has a name, and that name is the HCF.
💡 Here's the trick
Look at the diagram above. The HCF is the biggest number that divides every number in the set exactly. Because it has to divide the smallest number too, it can never be larger than that number.
🪄 How to do it
- 1.Write each number as a product of primes, as the diagram above shows.
- 2.Keep only the primes that appear in every number.
- 3.Take each of those at its lowest power and multiply.
✍️ Try it
Find the HCF of 18 and 27. Then: find the HCF of 16 and 40.
18 is 2 × 3² and 27 is 3³, so the shared part is 3², giving 9. 16 is 2⁴ and 40 is 2³ × 5, so the shared part is 2³, giving 8.
⚠️ Watch out
You might be thinking the HCF is always one of the two numbers. That happens only when one of them divides the other. For 18 and 27 the HCF is 9, which is neither of them.
🌟 Why this matters
Cutting cloth with nothing wasted, or seating guests in equal rows, both come down to the HCF. It is also how a fraction drops to its simplest form in one move.
Exam tip: Check that your answer divides both numbers, because a wrong HCF usually fails one of them.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Find the HCF of 30 and 42.”
- 1
Prime factorise each
30 = 2 × 3 × 5. 42 = 2 × 3 × 7. 🧲
- 2
Pick EACH common prime
Both have 2 and 3.
- 3
Multiply the shared primes
2 × 3 = 6
- 4
HCF ✓
HCF(30, 42) = 6
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
List factors of both, circle the common ones, keep the LARGEST.
HCF OF 16 & 24 · ROUND 1 OF 3
Quick check
✅ VERIFICATION
What is the HCF of 21 and 35?
⚡ Finish the activity above first to answer.
Keep going!
Up next: LCM
lcm of 4 and 6 = 12. hcf × lcm = a × b.