Before this page: be able to divide and see whether anything is left over from short division, and recognise the square numbers from recognising powers of a number.
Every factor has a partner
Suppose you have 24 counters. How many ways can you arrange them in equal rows, without leaving any counters out?
Four rows of 6 work, because 4 × 6 = 24. We call 4 and 6 a factor pair. Each is a factor of 24: a positive whole number that divides 24 exactly, with no remainder.
Finding one factor gives us its partner too. For example, 24 ÷ 3 = 8, so 3 and 8 form a pair. But 24 ÷ 5 leaves a remainder, so 5 is not a factor.
To find all the pairs, start at 1 and test the whole numbers in order. Write down both factors whenever the division is exact.
List every factor of 24
The numbers 1 to 24
Trials start at 1.
How do we know we've found them all?
We've found 1 × 24, 2 × 12, 3 × 8 and 4 × 6. Testing 5 gives a remainder. At 6, we get 6 × 4: a pair we already have, with its factors swapped.
As the first factor gets larger, its partner gets smaller. Once they cross, we can stop: any pair after that would repeat one we found earlier.
Stop below the root of the next square above the number
TipWe can find a stopping point before testing any factors. Since 5 × 5 = 25 is greater than 24, two factors of 24 cannot both be 5 or more. Every pair must therefore include 1, 2, 3 or 4. Testing those four numbers finds every pair.
Example: write down all the factors of 45
Work up from 1, writing both ends of each pair.
- 7 × 7 = 49 The next square above 45 is 49, so the trials stop below its root of 7.
- 1 × 45 = 45 1 divides every number, and its partner is the number itself.
- 3 × 15 = 45 2 leaves a remainder, so it joins no pair.
- 5 × 9 = 45 4 and 6 leave remainders too, and the six trials are done.
- 1, 3, 5, 9, 15, 45 Six factors, read off the two ends of three pairs.
When a factor is its own partner
What happens if the two factors meet exactly? With 36, we reach 6 × 6. This arrangement is a square, which is why we call 36 a square number.
List every factor of 36
The numbers 1 to 36
Trials start at 1.
Write 6 only once in the factor list, even though it appears twice in 6 × 6. That gives 36 nine factors. Every positive square number has an odd number of positive factors: the others come in pairs, with one unpaired factor in the middle.
Common mistakes
- Losing the partner of a number already tried Trying numbers in no order drops one end of a pair: 5 is tried but 40 ÷ 5 = 8 is never written, and 1, 2, 4, 5, 10, 20, 40 is seven factors where 40 has eight.
- Leaving out the widest pair 45 listed as 3, 5, 9, 15 has four factors, not six: 1 divides every number and every number divides itself, so 1 × 45 is the first pair.
- Counting a square number’s own partner twice 49 has three factors, not four. One place on the line stands for both ends of 7 × 7, so the 7 is written once: 1, 7, 49.
Summary
- A factor divides a number exactly, and the division gives its partner.
- Working up from 1 fills the list from both ends, so nothing is missed or written twice.
- To know where to stop, find the next square above the number: the trials stop below its root.
- 24 has eight factors: 1, 2, 3, 4, 6, 8, 12, 24.
- A square number has one factor that is its own partner, so it has an odd number of factors.