Concept 3 of 4
HCF & LCM: Family Reunions
Today's big idea
HCF is the biggest factor two numbers share, LCM the smallest multiple.
10 and 15 share 5 as their biggest factor, and first meet again at 30.
How it works
👀 What's going on?
Two numbers can share factors and can share multiples. The biggest factor they share is the HCF, and the smallest multiple they share is the LCM. The arithmetic is never the hard part. Knowing which one a question wants is.
💡 Here's the trick
Look at the two lists above. On the top row, the factors of 12 and 18 both contain 6, and 6 is the biggest they share, so HCF(12, 18) = 6. On the bottom row, the multiples of 4 and 6 first meet at 12, so LCM(4, 6) = 12. An HCF is never bigger than the numbers you started with. An LCM is never smaller.
🪄 How to do it
- 1.Decide whether the story splits things up or repeats them, the two rows in the picture above.
- 2.Splitting into equal groups wants the HCF, so list factors as the picture above does and take the biggest shared one.
- 3.Something coming round together again wants the LCM, so list multiples and take the first shared one.
✍️ Try it
You have 12 red and 18 blue pens to split into identical bundles with none left over. Then two lights flash every 4 and every 6 seconds. Which story needs the HCF?
The pens are being split, so that one is the HCF, and the biggest bundle size dividing both 12 and 18 is 6. The lights repeat until they coincide, so that one is the LCM, and the first shared multiple of 4 and 6 is 12 seconds. Splitting made the answer smaller than the numbers, repeating made it bigger.
⚠️ Watch out
You might be thinking certain words settle it, so that maximum means HCF and smallest means LCM. That holds until a question asks for the smallest number of bundles, which is still an HCF problem. Ask whether the story splits or repeats, then check the size: a splitting answer comes out below your numbers, a repeating answer above them.
🌟 Why this matters
Sharing two different bags of sweets fairly between friends is an HCF question. Working out when two buses running every 15 and 20 minutes next leave together is an LCM question. Both show up whenever things have to line up.
Exam tip: Sense-check the size every time, an HCF no bigger than either number and an LCM no smaller.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Find HCF(20, 30) and LCM(4, 10).”
- 1
Factors of each (HCF)
Factors 20: 1,2,4,5,10,20. Factors 30: 1,2,3,5,6,10,15,30.
- 2
Shared factors
Both lists share 1, 2, 5, 10. Biggest = 10. So HCF = 10. 🤝
- 3
Multiples (LCM)
Multiples 4: 4, 8, 12, 16, 20… Multiples 10: 10, 20…
- 4
First shared multiple
Smallest in both = 20.
- 5
Answer ✓
HCF(20,30) = 10 · LCM(4,10) = 20
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
HCF = biggest SHARED factor. LCM = smallest SHARED multiple.
HCF HUNT
8 & 12
Tap every number that appears in BOTH lists. (3 to find.)
Factors of 8
Factors of 12
Shared found: 0 of 3
Quick check
✅ VERIFICATION
Forks come in packs of 8, spoons in packs of 12. What is the smallest number of each you can buy to end up with equal numbers?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Quick Tests
Answer 'does 3 go into it?' in about a second.