The idea, step by step
Two factors make one product
A factor is a whole number that divides another whole number exactly, with no remainder. Because 3 × 8 = 24, both three and eight are factors of twenty-four. They form a factor pair.
An array shows the same relationship. Twenty-four counters can be arranged in three rows of eight. Turning the array makes eight rows of three; this uses the same factor pair in a different order.
| Rows | Counters in each row | Total |
|---|---|---|
| 1 | 24 | 24 |
| 2 | 12 | 24 |
| 3 | 8 | 24 |
| 4 | 6 | 24 |
Keep a list so no pair is missed
Begin with one and the number itself. Try two, then three, then four, recording only divisions with no remainder. For 24, the pairs are 1 and 24, 2 and 12, 3 and 8, and 4 and 6. Five does not work. Six would repeat the pair 4 and 6.
Now collect the numbers from the pairs and put them in order. The positive factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Include one and the number itself; do not include zero because division by zero is not defined.
A common trap
Listing only the first number in each factor pair.
Instead: Both numbers divide the target exactly. From 3 × 8 = 24, record three and eight, not just three.
See the thinking
Find the pairs for eighteen
Find all positive factor pairs of 18.
- One times eighteen gives eighteen.
- Two times nine and three times six also give eighteen.
- Four and five do not divide eighteen exactly. Six would repeat the pair three and six.
Choose a rectangular seating plan
Twenty counters must form a rectangle with at least two rows and at least two counters in each row. Give the possible factor pairs.
- The pairs for twenty are 1 × 20, 2 × 10 and 4 × 5.
- The one-row arrangement does not meet the condition.
- The remaining pairs give two different shapes, ignoring turns.
Check your understanding
Is 7 a factor of 28? Give a multiplication fact.
Show answer
Yes: 7 × 4 = 28.
Twenty-eight can be divided into seven equal groups of four, with no remainder.
Your turn
6 questions · 11 marks · Work at your own pace.
Read the questions online
- List all positive factor pairs of 12.2 marks
- List the positive factors of 15 in order.2 marks
- Is 6 a factor of 42? Explain.1 mark
- Is 5 a factor of 32? Explain.1 mark
- List each factor pair of 36 once, including the equal-factor pair.2 marks
- A garden has 30 square tiles. It must have at least three rows and at least three tiles per row. Give all possible rectangular factor pairs, ignoring turns.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- List all positive factor pairs of 12.2 marks
Try one, two and three. Four would repeat three and four.
1 × 12, 2 × 6 and 3 × 4. - List the positive factors of 15 in order.2 marks
Use 1 × 15 and 3 × 5.
1, 3, 5, 15. - Is 6 a factor of 42? Explain.1 mark
6 × 7 = 42, so division by six leaves no remainder.
Yes. - Is 5 a factor of 32? Explain.1 mark
5 × 6 = 30 and 5 × 7 = 35. Neither is thirty-two.
No. - List each factor pair of 36 once, including the equal-factor pair.2 marks
Five does not divide thirty-six. The pair six and six is included once.
1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. - A garden has 30 square tiles. It must have at least three rows and at least three tiles per row. Give all possible rectangular factor pairs, ignoring turns.3 marks
The pairs are 1 × 30, 2 × 15, 3 × 10 and 5 × 6.
Remove pairs with a side shorter than three.
3 × 10 and 5 × 6.
Curriculum reference & lesson scope
Australian Curriculum v9.0; SA Curriculum Mathematics 2026
Factor pairs; multiples; divisibility; factor/divisibility algorithms.
Official curriculum source ↗