Factors, multiples, primes, squares and cubes
These number properties come up again and again in the SATs reasoning papers, often in puzzle form. They are easy to confuse, so clear definitions and a few reliable methods make a big difference.
Factors and multiples
A factor divides into a number exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple is in a number's times table: the multiples of 6 are 6, 12, 18, 24 and so on. A simple way to keep them apart: factors fit into a number; multiples come out of it.
The reliable way to find all the factors of a number is to list them in pairs, starting from 1 and stopping when the pairs meet:
- 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Working in pairs means no factor gets missed. Year 6 pupils also find common factors (factors shared by two numbers) and common multiples (multiples shared by two numbers). Listing both sets side by side and circling the matches is the clearest method.
Prime numbers
A prime number has exactly two factors: 1 and itself. The primes under 20 are 2, 3, 5, 7, 11, 13, 17 and 19. Year 6 pupils should know the primes up to 19 by heart and be able to check whether a number up to 100 is prime.
- 1 is not primeIt has only one factor.
- 2 is primeThe only even prime number.
- 9 is prime because it's odd9 = 3 × 3. Not all odd numbers are prime.
To check a number up to 100, try dividing by 2, 3, 5 and 7. If none of them divide exactly, it's prime. A composite number (not prime) can be broken into prime factors: 60 = 2 × 2 × 3 × 5.
Square and cube numbers
A square number is a number multiplied by itself: 5² = 5 × 5 = 25. A cube number is a number multiplied by itself three times: 2³ = 2 × 2 × 2 = 8. The names come from shapes: 25 counters make a 5 by 5 square, and 8 cubes make a 2 by 2 by 2 cube.
- 5² = 5 × 5 = 25Squared means multiplied by itself.
- 5² = 5 × 2 = 10The most common error with indices.
- 3³ = 3 × 3 × 3 = 27Not 3 × 3 = 9.
Children should know the square numbers to 12² = 144 and the cube numbers 1, 8, 27, 64 and 125.
In the SATs
These properties appear mostly on the reasoning papers, often as puzzles: “I am a two-digit square number and a multiple of 3. What could I be?” Questions also ask children to list factors, identify primes in a set, find a common multiple, or explain why a number is or isn't prime. The arithmetic paper sometimes includes squared and cubed numbers within calculations.
Common mistakes
- Confusing factors and multiples.
- Thinking 1 is prime, or that all odd numbers are prime.
- Reading 5² as 5 × 2.
- Missing factors because they weren't found in pairs.
- Stopping after one common multiple when the question asks for the lowest.
Teaching ideas
Rectangle arrays
Give children 24 counters and ask them to make every possible rectangle. Each rectangle shows a factor pair. Primes are the numbers that only make one long line.
Sieve of Eratosthenes
On a 100 square, cross out multiples of 2, 3, 5 and 7 (except the numbers themselves). What's left are the primes. Children see why checking up to 7 is enough for numbers under 100.
Build the squares and cubes
Use multilink cubes to build 2³, 3³ and 4³. Seeing 64 cubes in a 4 by 4 by 4 block makes the difference between squaring and cubing unforgettable.
Who am I?
Set number riddles that combine properties: “I'm a prime between 30 and 40, and my digits add to 10.” Pupils then write their own for a partner.
Try it: five quick questions
SATs-style questions to use as a starter or a quick check. Tap to see each answer.
List the factors of 30.
Show answer
1, 2, 3, 5, 6, 10, 15, 30
Find the lowest common multiple of 4 and 6.
Show answer
12
Is 51 prime? Explain.
Show answer
No: 51 = 3 × 17.
Calculate 4³.
Show answer
64
Which number is both a square number and a cube number, and less than 100 (apart from 1)?
Show answer
64: 8² and 4³.
Free worksheets for this topic
Printable A4 PDFs with answers. Free to use in class or at home.
