Calculation
Factors, Multiples and Prime Numbers
Year 3 · Year 4 · Year 5 · Year 6
- ✓Identify factors of a given whole number.
- ✓Identify multiples of a given whole number.
- ✓Find common factors and common multiples of two numbers.
- ✓Recognise and identify prime numbers up to 100.
- ✓Calculate and use the notation for square numbers and cube numbers.
Key concepts
Factors are whole numbers that divide exactly into another number, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides into 12 perfectly.
Multiples are the results you get when you multiply a number by other whole numbers (like in its times table). For example, the multiples of 5 are 5, 10, 15, 20, 25, and so on.
Common factors are factors that two or more numbers share. For example, the factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 18 are 1, 2, 3, 6, 9, 18. The common factors of 12 and 18 are 1, 2, 3, and 6.
Common multiples are multiples that two or more numbers share. For example, the multiples of 2 are 2, 4, 6, 8, 10, 12... and the multiples of 3 are 3, 6, 9, 12, 15... The common multiples of 2 and 3 include 6, 12, 18, and so on.
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. For example, 7 is a prime number because its only factors are 1 and 7. The number 1 is not a prime number.
A square number is the result of multiplying a whole number by itself. We use a small '2' (squared) to show this. For example, 3 squared (3²) means 3 × 3 = 9. So, 9 is a square number.
A cube number is the result of multiplying a whole number by itself, and then by itself again. We use a small '3' (cubed) to show this. For example, 2 cubed (2³) means 2 × 2 × 2 = 8. So, 8 is a cube number.
Key facts to remember
- 11 is a factor of every whole number.
- 2Every whole number is a multiple of itself.
- 3Prime numbers have exactly two factors: 1 and themselves.
- 4The only even prime number is 2.
- 5The number 1 is not a prime number.
- 6Square numbers are written with a small '2' (e.g., 4²).
- 7Cube numbers are written with a small '3' (e.g., 4³).
- 8To find factors, think of division. To find multiples, think of multiplication (times tables).
Worked examples
Example 1
a) List all the factors of 24. b) List the first five multiples of 7.
Answer
a) Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. b) First five multiples of 7: 7, 14, 21, 28, 35.
Always remember to include 1 and the number itself when listing factors.
Example 2
a) Find the common factors of 10 and 15. b) Find the first two common multiples of 4 and 6.
Answer
a) Common factors of 10 and 15: 1, 5. b) First two common multiples of 4 and 6: 12, 24.
You may need to list several multiples to find the common ones.
Example 3
a) Is 13 a prime number? Explain why. b) Calculate 5² and 3³.
Answer
a) Yes, 13 is a prime number because its only factors are 1 and 13. b) 5² = 25, 3³ = 27.
Remember that 1 is not a prime number.
Common mistakes
- ✗Confusing factors (numbers that divide into) with multiples (numbers in the times table of).
- ✗Thinking that 1 is a prime number (it only has one factor, not two).
- ✗Forgetting to include 1 and the number itself when listing all factors.
- ✗Calculating a square number like 4² as 4 × 2 = 8, instead of 4 × 4 = 16.
- ✗Calculating a cube number like 2³ as 2 × 3 = 6, instead of 2 × 2 × 2 = 8.
Exam tips
- ★When listing factors, always start with 1 and the number itself, then work inwards in pairs (e.g., 1 x 24, 2 x 12, 3 x 8, 4 x 6).
- ★To find common multiples, list the first few multiples of each number until you find numbers that appear in both lists.
- ★To check if a number is prime, try dividing it by small prime numbers (2, 3, 5, 7...). If none divide it exactly, it's likely prime.
- ★Practise recalling the first few square numbers (1, 4, 9, 16, 25, 36...) and cube numbers (1, 8, 27, 64, 125...).
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