Factor Pairs – Every Factor Has a Partner
When you divide a number by one of its factors, the answer is also a factor. These two factors that multiply together to give the original number are called a factor pair.
What Is a Factor Pair?
A factor pair of N is two whole numbers (a, b) where a × b = N.
Finding All Factor Pairs
Factor pairs of 24
| Factor a | Factor b | Check: a × b |
|---|---|---|
| 1 | 24 | 24 ✓ |
| 2 | 12 | 24 ✓ |
| 3 | 8 | 24 ✓ |
| 4 | 6 | 24 ✓ |
√24 ≈ 4.9, so stop at 4. Factor pairs: (1,24), (2,12), (3,8), (4,6)
Factor pairs of 36
| Factor a | Factor b | Check |
|---|---|---|
| 1 | 36 | ✓ |
| 2 | 18 | ✓ |
| 3 | 12 | ✓ |
| 4 | 9 | ✓ |
| 6 | 6 | ✓ (perfect square pair) |
Perfect Squares Have an Odd Factor
When a = b in a factor pair (like 6 × 6 = 36), the number is a perfect square. This gives it an odd number of factors because one factor (the square root) has no separate partner.
| Number | Perfect Square? | Number of Factors |
|---|---|---|
| 16 | Yes (4×4) | 5 (1, 2, 4, 8, 16) |
| 12 | No | 6 (1, 2, 3, 4, 6, 12) |
| 25 | Yes (5×5) | 3 (1, 5, 25) |
Key Takeaways
- A factor pair multiplies to give the original number.
- Test divisors up to the square root; each gives a pair.
- Perfect squares have one unpaired factor (the square root itself).
- Factor pairs are especially useful when building factor trees.
Practice Questions
- List all factor pairs of 30.
- List all factor pairs of 49. Is 49 a perfect square?
- A rectangle has area 48 cm². List all possible whole-number dimensions.
- How many factor pairs does 100 have?
- Find a number between 50 and 60 that has exactly 3 factor pairs.