Concept 1 of 4
Factors & Multiples: The Number Families
Today's big idea
Factors divide inwards and run out; multiples multiply outwards and never do.
Factors of 15 stop at {1, 3, 5, 15}, but its multiples run on for ever.
How it works
👀 What's going on?
Every number belongs to two families. One is the numbers that fit into it exactly, the other is the numbers it fits into. They run in opposite directions, which is where nearly all the confusion comes from.
💡 Here's the trick
Look at the 12 in the middle above. Its factors run inwards: 1, 2, 3, 4, 6 and 12, pairing up as 1x12, 2x6 and 3x4. Its multiples run outwards: 12, 24, 36, 48 and onwards forever. Factors are small and the list ends. Multiples are big and the list never does.
🪄 How to do it
- 1.To list factors, work upwards from 1 as the picture above does, checking what divides evenly.
- 2.Write each one down as a pair, and stop once the pairs start meeting in the middle.
- 3.To list multiples, just count up in that number: one times it, two times it, three times it.
✍️ Try it
Find every factor of 18. Then find the first multiple that 4 and 6 both share.
The factor pairs of 18 are 1x18, 2x9 and 3x6, giving {1, 2, 3, 6, 9, 18}, six of them. For the shared multiple, 4 goes 4, 8, 12 while 6 goes 6, 12, so the first they share is 12.
⚠️ Watch out
You might be thinking a number has roughly as many multiples as factors. It helps to remember that factors are few and multiples are many. 18 has exactly six factors and that list is finished. Its multiples run on forever. If your factor list is growing past a dozen, you are probably listing multiples by mistake.
🌟 Why this matters
Factors are how you cut a cake into equal pieces or set out chairs in equal rows. Multiples are how you work out when two things will happen together again, like two buses leaving at the same moment.
Exam tip: Listing factors in pairs stops you missing one, which is the usual reason these get marked wrong.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Find all the factors of 20.”
- 1
Start at 1 and pair
Does 1 divide 20? Yes → pair 1 & 20.
- 2
Try 2
20 ÷ 2 = 10 ✓ → pair 2 & 10.
- 3
Try 3 (no), then 4
20 ÷ 4 = 5 ✓ → pair 4 & 5. Next is 5 — already found. Stop.
- 4
List all factors ✓
Factors of 20: {1, 2, 4, 5, 10, 20}
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
FACTORS divide IN evenly. MULTIPLES multiply OUT by 1, 2, 3...
FACTOR HUNT
18
Tap every number that divides 18 evenly.
Found 0 of 8 factors
Quick check
✅ VERIFICATION
Which number is a factor of 24 but NOT a multiple of 3?
⚡ Finish the activity above first to answer.
Keep going!
Up next: The Roots of Numbers
Spot primes vs composites.