The idea, step by step
Look for factor pairs
A factor divides a whole number exactly. List factor pairs systematically: start with 1 and the number, then test 2, 3 and so on. A prime number is greater than 1 and has exactly two positive factors, 1 and itself. A composite number is a whole number greater than 1 with more than two positive factors.
The numbers 0 and 1 are neither prime nor composite. The number 1 has only one positive factor; 0 does not meet either definition. The number 2 is prime even though it is even. Every other even number greater than 2 has 2 as an extra factor.
Recognise squares
A square number can be arranged as a square array. For example, 7 rows of 7 make 49. The notation means , not .
One number may have several descriptions. The number 36 is both composite and square. These descriptions are not three separate boxes into which each number must fit.
A common trap
Every odd number is prime.
Instead: Odd numbers may have factors other than 1 and themselves. For example, 21=3×7 is composite.
See the thinking
Worked example 1
Is 29 prime or composite?
- Its first factor pair is 1 and 29.
- None of 2, 3, 4 or 5 divides 29 exactly. The next candidate, 6, has a square greater than 29, so no smaller factor pair remains.
Worked example 2
Classify 36 and explain whether it is square.
- Its pairs are 1×36, 2×18, 3×12, 4×9 and 6×6.
- More than two factors make it composite; the equal pair 6×6 makes it square.
Check your understanding
Is 1 prime? Explain.
Show answer
No; it is neither prime nor composite.
A prime needs exactly two positive factors. The number 1 has only one.
Your turn
6 questions · 10 marks · Work at your own pace.
Read the questions online
- List all positive factors of 18.2 marks
- Classify 17 as prime or composite.1 mark
- Explain why 27 is composite.2 marks
- List the square numbers from 1 to 50.2 marks
- A garden has 8 rows of 8 plants. How many plants?1 mark
- Sam says every square number is composite. Give a counterexample.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- List all positive factors of 18.2 marks
Use pairs 1×18, 2×9 and 3×6.
1, 2, 3, 6, 9, 18. - Classify 17 as prime or composite.1 mark
Its only positive factors are 1 and 17.
Prime. - Explain why 27 is composite.2 marks
The pair 3×9 supplies extra factors.
It has factors 3 and 9 as well as 1 and 27. - List the square numbers from 1 to 50.2 marks
Calculate 1×1 through 7×7; 8×8=64 exceeds 50.
1, 4, 9, 16, 25, 36, 49. - A garden has 8 rows of 8 plants. How many plants?1 mark
8×8=64.
64 plants. - Sam says every square number is composite. Give a counterexample.2 marks
1=1×1 and has only one positive factor.
1 is square but is not composite.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics
Prime/composite/square numbers; factors/multiples revision; efficient calculation and estimation.
Official curriculum source ↗