Two related ideas
Factors and multiples are two sides of multiplication. Look at 3 × 4 = 12:
- 3 and 4 are factors of 12.
- 12 is a multiple of 3 and of 4.
| Idea | Question | For 12 |
|---|---|---|
| Factor | What divides into it exactly? | 1, 2, 3, 4, 6, 12 |
| Multiple | What do I get when I multiply it? | 12, 24, 36, 48, … |
A number has a limited list of factors, but its multiples go on forever.
Finding factors with factor pairs
A factor pair is two numbers whose product is the number. Start at 1 and work up:
Factors of 36
| Try | Divisible? | Pair |
|---|---|---|
| 1 | yes | 1 × 36 |
| 2 | yes | 2 × 18 |
| 3 | yes | 3 × 12 |
| 4 | yes | 4 × 9 |
| 5 | no | |
| 6 | yes | 6 × 6 |
When the pairs start repeating (or meet in the middle), stop. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
Square numbers like 36 have an odd number of factors, because one pair uses the same number twice.
Finding multiples
Skip count, or multiply by 1, 2, 3, …
- Multiples of 6: 6, 12, 18, 24, 30, 36, …
- Multiples of 8: 8, 16, 24, 32, 40, …
Every number is a multiple of itself (× 1) and a factor of itself.
Common factors and common multiples
- A common factor divides two numbers. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Common factors: 1, 2, 3, 6.
- A common multiple appears in both lists of multiples. Multiples of 6 and 8 both include 24, 48, 72, …
These lead to the greatest common factor and least common multiple, used with fractions.
Real-life uses
- Arranging 24 chairs in equal rows: the rows must be a factor of 24 (1, 2, 3, 4, 6, 8, 12, 24).
- Hot dogs come in packs of 10, buns in packs of 8. Common multiples tell you when the numbers match (40, 80, …).
Worked examples
Example 1: List the factors of 12.
Pairs: 1 × 12, 2 × 6, 3 × 4. Factors: 1, 2, 3, 4, 6, 12.
Example 2: Is 45 a multiple of 9?
9 × 5 = 45. Yes.
Example 3: List the first five multiples of 7.
7, 14, 21, 28, 35.
Common mistakes
- Mixing up factors and multiples. Factors are smaller (or equal); multiples are bigger (or equal).
- Forgetting 1 and the number itself in a factor list.
- Missing a factor pair by skipping numbers. Test every number in order.
Try it yourself
- List the factors of 20.
- List the first six multiples of 4.
- Find the common factors of 16 and 24.
- Is 8 a factor or a multiple of 32?
Answers: 1) 1, 2, 4, 5, 10, 20 2) 4, 8, 12, 16, 20, 24 3) 1, 2, 4, 8 4) factor
Quick recap
- Factors divide a number exactly; multiples are what you get by multiplying it.
- Find factors in factor pairs.
- Common factors and common multiples are shared by two numbers.