This Year 5 factors lesson is written for pupils aged 9-10 and follows the National Curriculum. In short, it shows you how to find factors and multiples, identify primes up to 100, and use square and cube notation. Before you start, it helps to revise place value to 1 000 000, because this lesson builds on it.
Year 5 factors lesson: what you will learn
Hello, I am Otto! Numbers have secrets. Some numbers can be shared out equally in many ways, like 24 sweets. Others, like 13, can only be shared one way. In this chapter you will meet factors, multiples, prime numbers, composite numbers, square numbers and cube numbers. These ideas help with fractions, with times tables and with solving puzzles.
Starter activity
Make 12 counters into a rectangle. You can make a rectangle with 1 row of 12, 2 rows of 6 or 3 rows of 4. Now try 13 counters. You can only make 1 row of 13. Keep thinking about why 12 and 13 behave so differently as you read on.
This array has 3 rows of 4 dots, so it shows 3 × 4 = 12.
Factors and factor pairs
Definition: factor
A factor of a number is a whole number that divides it exactly, leaving no remainder. 6 is a factor of 24 because 24 ÷ 6 = 4. Every number has 1 and itself as factors.
Method: listing all the factors
- Start with the pair 1 and the number itself.
- Test 2, then 3, then 4 and so on. When a number divides exactly, write down the pair, for example 3 and 8 for 24.
- Stop when you reach a factor that you have already written down as the partner of a smaller one.
- Write the factors in order, smallest first.
Example 1
List all the factors of 36 and write its factor pairs.
1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. The factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. The pair 6 × 6 gives the factor 6 only once, which is why 36 has an odd number of factors.
Key fact: divisibility tests
A number is divisible by 2 if it ends in 0, 2, 4, 6 or 8. It is divisible by 5 if it ends in 0 or 5, and by 10 if it ends in 0. It is divisible by 3 if its digits add to a multiple of 3, and by 9 if the digits add to a multiple of 9. For example 4,572 has digit sum 4 + 5 + 7 + 2 = 18, so it is divisible by both 3 and 9.
Multiples and common multiples
Definition: multiple
A multiple of a number is a number in its times table. The multiples of 7 are 7, 14, 21, 28, 35 and so on. A multiple is made by multiplying the number by 1, 2, 3, 4 and so on.
Definition: common factor and common multiple
A common factor of two numbers divides them both. A common multiple is in both times tables. The highest common factor is the biggest common factor and the lowest common multiple is the smallest common multiple.
Example 2
Find the lowest common multiple of 6 and 8, and the highest common factor of 18 and 30.
Multiples of 6: 6, 12, 18, 24, 30. Multiples of 8: 8, 16, 24, 32. The first number in both lists is 24, so the lowest common multiple is 24. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. The common factors are 1, 2, 3 and 6, so the highest common factor is 6.
Watch out
Factors are never bigger than the number, but multiples are never smaller. Many pupils mix up the words. Remember: factors are found by dividing and are few in number, multiples are found by multiplying and go on for ever.
Prime and composite numbers
Definition: prime number
A prime number has exactly two factors, 1 and itself. The prime numbers up to 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. A number with more than two factors is called a composite number. The number 1 is neither prime nor composite, because it has only one factor.
Key fact: facts about primes
2 is the only even prime number. Every other even number has the factor 2. No prime number bigger than 5 ends in 0, 2, 4, 5, 6 or 8. To test if a number up to 100 is prime, try dividing it by 2, 3, 5 and 7. If none of them works, the number is prime. Beware: 91 = 7 × 13 is composite even though it looks like a prime.
Definition: prime factor
A prime factor is a factor that is also a prime number. Every composite number can be written as a product of its prime factors in exactly one way, apart from the order. For example 60 = 2 × 2 × 3 × 5.
Method: a factor tree
- Write the number and split it into any two factors.
- Circle any factor that is prime and split any factor that is not.
- Keep going until every branch ends in a prime.
- Write the primes in order as a product. Use powers if you like: 60 = 2² × 3 × 5.
Example 3
Write 72 as a product of prime factors.
72 = 8 × 9. Then 8 = 2 × 2 × 2 and 9 = 3 × 3. So 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
Square numbers and cube numbers
A 4 by 4 array of tiles has 16 tiles. 16 is a square number.
Definition: square number
A square number is a number multiplied by itself. We write it with a small 2: 4² = 4 × 4 = 16. The first twelve squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. A square number has an odd number of factors.
Definition: cube number
A cube number is a number multiplied by itself and then by itself again. We write it with a small 3: 3³ = 3 × 3 × 3 = 27. The first five cubes are 1, 8, 27, 64 and 125, and 10³ = 1,000.
Example 4
Work out 5² and 5³, and say whether 50 is a square number.
5² = 5 × 5 = 25. 5³ = 5 × 5 × 5 = 25 × 5 = 125. For 50: 7² = 49 and 8² = 64, so 50 lies between two squares and is not a square number.
Watch out
5² does NOT mean 5 × 2. It means 5 × 5. In the same way 5³ is not 5 × 3. Check by working out 2² = 4 and 2 × 2 = 4, but 3² = 9 while 3 × 2 = 6.
Example 5
A hall has 144 chairs in a square arrangement. How many chairs are in each row?
The rows and columns are equal, so the number in each row is the square root of 144. Since 12 × 12 = 144, there are 12 chairs in each row.
Did you know?
The ancient Greeks called numbers like 1, 4, 9 and 16 "square" because they thought of them as dots in a square pattern. About 2,300 years ago a Greek mathematician called Eratosthenes invented a clever way to find primes. You write the numbers on a grid and cross out multiples of 2, 3, 5 and 7. The numbers left are primes. It is called the Sieve of Eratosthenes.
Otto’s tip
To check whether a number is prime, divide it by 2, 3, 5 and 7 only. For numbers under 100, if none of them divide exactly, the number is prime. Learn the primes up to 20 by heart, they appear in lots of questions!
How to use this Year 5 factors lesson
First, read the key points so that you know where the Year 5 factors lesson is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the factors, multiples, primes, squares and cubes worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the factors, multiples, primes, squares and cubes test and mark it with the test answers.
Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 5 factors lesson gives you everything you need for the chapter.
Similar pages
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AnswersFactors, multiples, primes, squares and cubes – Year 5 maths worksheet answers to download as PDF
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QuizFactors, multiples, primes, squares and cubes – Year 5 maths quiz to download as PDF
TestFactors, multiples, primes, squares and cubes – Year 5 maths test to download as PDF
Test answersFactors, multiples, primes, squares and cubes – Year 5 maths test answers to download as PDF
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