Year 5 maths · Key Stage 2 · Number – multiplication and division

Year 5 factors, multiples, primes, squares and cubes

In Year 5 children study the structure of whole numbers: which numbers divide into others (factors), which numbers appear in times tables (multiples), which have only two factors (prime numbers), and which are made by multiplying a number by itself (squares) and then by itself again (cubes). These ideas underpin fractions, ratio and algebra later on.

Factors, multiples, primes, squares and cubes, explained

A factor of a number is a whole number that divides into it exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12. Factors come in pairs that multiply to make the number, so 12 has the factor pairs 1 × 12, 2 × 6 and 3 × 4. Finding all of them in an orderly way, starting from 1 and working upwards until the pairs meet, means none are missed. A multiple is a number in a times table, such as 12, 24 and 36 for the 12 times table. Children also find common factors, the factors that two numbers share. The factors of 12 are 1, 2, 3, 4, 6 and 12, and the factors of 18 are 1, 2, 3, 6, 9 and 18, so the common factors of 12 and 18 are 1, 2, 3 and 6.

A prime number has exactly two factors: 1 and itself. The primes up to 19 are 2, 3, 5, 7, 11, 13, 17 and 19, and children are expected to recall them. A number with more than two factors is a composite number. The number 1 is neither prime nor composite, as it has only one factor. And 2 is the only even prime. Children test whether a number up to 100 is prime by checking whether it can be divided by 2, 3, 5 or 7: any composite number up to 100 has at least one of these as a factor. The prime factors of a number are the factors that are themselves prime: the prime factors of 36 are 2 and 3.

A square number is what you get when you multiply a whole number by itself, written with a small raised 2: 6² = 6 × 6 = 36. They get their name from the shape: 36 counters can be arranged into a square, 6 by 6. A cube number multiplies a whole number by itself and then by itself again, written with a raised 3: 4³ = 4 × 4 × 4 = 64, which is the number of small cubes in a 4 by 4 by 4 cube. Children learn to recognise square numbers (1, 4, 9, 16, 25 and so on, with many schools working up to 12² = 144, which links to the times tables) and cube numbers (1, 8, 27, 64, 125, and 10³ = 1,000).

Children put all of this to work in problems. They use factors to make a calculation easier, for example 4 × 35 = 2 × 2 × 35 = 70 × 2 = 140. They might write 3 × 270 as 3 × 3 × 9 × 10 = 9² × 10 = 810. They also use multiples and factors to solve problems, such as deciding whether 96 children can be split into equal teams of 6, 7 or 8 with nobody left over, or listing all the ways 24 sweets can be shared equally.

Words your child will hear

Factor: A whole number that divides exactly into another. 4 is a factor of 36.
Multiple: The result of multiplying a number by a whole number. 36 is a multiple of 4.
Prime number: A number with exactly two factors, 1 and itself, such as 13.
Composite number: A number with more than two factors, such as 15 or 36.
Square number: The answer when a whole number is multiplied by itself, such as 25 = 5 × 5 = 5².
Cube number: The answer when a whole number is multiplied by itself and then by itself again, such as 27 = 3 × 3 × 3 = 3³.
Prime factor: A factor that is itself a prime number. The prime factors of 12 are 2 and 3.

What the National Curriculum says

In Year 5, pupils should be taught to:

Identify multiples and factors, including finding all factor pairs of a number, and common factors of 2 numbers
Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers
Establish whether a number up to 100 is prime and recall prime numbers up to 19
Recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)
Solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Find all the factor pairs of 36. Is it prime? Is it a square number? What are its common factors with 24?

Work through the whole numbers in order, checking whether each divides exactly. 1 does (1 × 36). 2 does (2 × 18). 3 does (3 × 12). 4 does (4 × 9). 5 does not. 6 does (6 × 6). The next number to try would be 7, but the pairs have now met, so stop.
```
1 × 36
2 × 18
3 × 12
4 × 9
6 × 6
```
List every factor, in order: 1, 2, 3, 4, 6, 9, 12, 18 and 36. That is 9 factors. 6 only appears once because it is paired with itself.
36 has more than two factors, so it is not prime. It is a composite number. It is a square number, because 6 × 6 = 36, and so 36 = 6².
Now list the factors of 24: 1, 2, 3, 4, 6, 8, 12 and 24. Circle the ones that also appear in the list for 36.
```
36: 1 2 3 4 6 9 12 18 36
24: 1 2 3 4 6 8 12 24
```

Factor pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. 36 is not prime but is a square number. Its common factors with 24 are 1, 2, 3, 4, 6 and 12.

Common mistakes and how to help

Mixing up factors and multiples

The two words sound alike and both link two numbers by multiplication, so children use them interchangeably.

What helps: A number has only a few factors, and none is bigger than the number itself: they ‘fit into’ it. Multiples go on for ever, because they are the answers in a times table. List the factors of 12 and the first five multiples of 12 side by side.

Thinking 1 is a prime number, or that 2 is not prime

1 seems to fit the idea of ‘only divisible by 1 and itself’, and 2 is even, whereas most even numbers are not prime.

What helps: Count the factors: a prime has exactly two. 1 has one factor, so it is not prime. 2 has two, 1 and 2, so it is prime. It is the only even one.

Assuming numbers like 51 or 91 are prime

They look as if they cannot be divided by anything. Children check 2, 5 and perhaps 3, then stop.

What helps: Test 2, 3, 5 and 7 every time. 51 = 3 × 17 and 91 = 7 × 13. A number is divisible by 3 if its digits add to a multiple of 3: 5 + 1 = 6.

Reading 5² as 5 × 2 and 4³ as 4 × 3

The small raised number looks like a number to multiply by. The idea of repeated multiplication is new.

What helps: Say it as ‘five squared’ and write it out: 5 × 5 = 25. Build a square from 25 counters, then a 3 by 3 by 3 cube from small blocks to show 3³ = 27.

How to help at home

Arrange objects in arrays. Take 12, 18 or 24 buttons, coins or pasta shapes and arrange them in as many rectangles as possible. Each rectangle shows a factor pair. Then try 13: it only makes a single line, which shows it is prime.
Play ‘am I a multiple?’. Call out a number between 1 and 100 and ask whether it is a multiple of 3, 4 or 6, and how your child knows. Then ask for a factor of that number.
Hunt for primes. Write the numbers 1 to 50 on paper. Cross out 1, then every multiple of 2, 3, 5 and 7, but leave those four numbers themselves. What is left is the primes. Ask why 11 and 13 are still there.
Build squares and cubes. Use building blocks or small boxes to make squares of side 1, 2, 3 and 4, and count how many each takes. Then try a small cube and see how fast the numbers grow.

Builds on

Year 4: Times tables to 12 × 12

Leads on to

Year 6: Long multiplication and division
Year 6: Order of operations and multi-step problems