Multiples versus factors
Clarify two easily confused ideas using number lines and factor rainbows.
Quick answer
Multiples are what you get when you multiply a number by another number. Factors are the numbers you multiply together to get a product. For example, 12 is a multiple of 3 because 3 × 4 = 12, and 3 and 4 are factors of 12.
Learning outcome
By the end of this page, you will be able to tell the difference between multiples and factors and find all the factors of a number using a factor rainbow.
- I can list multiples of a number in order
- I can find all the factors of a number using pairs
- I can use a factor rainbow to show all the factor pairs
- I can explain the difference between multiples and factors
Why this matters
Understanding multiples and factors helps you solve puzzles, share things fairly, and work out how many groups you can make. These ideas are used in cooking, sports, games, and even in science when scientists need to understand patterns in nature.
Knowing your multiples and factors makes it easier to add and subtract fractions, find the lowest common multiple, and solve problems with division. It's like having a superpower for maths!
How it works
Let's break down what multiples and factors mean using simple steps:
- Multiples are the results when you multiply a number by 1, 2, 3, 4, and so on.
- Factors are the numbers you multiply together to make a product.
- For example, if you have 3 × 4 = 12, then 12 is a multiple of both 3 and 4.
- But 3 and 4 are also factors of 12 because they multiply to make 12.
- Think of it like a recipe: if you need 3 eggs for every 4 cups of flour, then 3 and 4 are factors of the total amount of ingredients.
Examples you can see
Let's look at the number 12 to understand multiples and factors.
Multiples of 12:
- 12 × 1 = 12
- 12 × 2 = 24
- 12 × 3 = 36
- 12 × 4 = 48
- 12 × 5 = 60
So the multiples of 12 are 12, 24, 36, 48, 60, and so on.
Factors of 12:
Now, let's find all the numbers that multiply together to make 12:
- 1 × 12 = 12
- 2 × 6 = 12
- 3 × 4 = 12
So the factors of 12 are 1, 2, 3, 4, 6, and 12.
Try this with your favourite number! What are the multiples of 5? What are the factors of 5?
Try it as a mini adventure
The Treasure Hunt Challenge
You are going on a treasure hunt in a magical garden. You need to collect 12 golden coins to unlock the treasure chest. But the coins are hidden in groups:
- How many groups of 3 coins can you make? (Answer: 4 groups)
- How many groups of 4 coins can you make? (Answer: 3 groups)
- How many groups of 2 coins can you make? (Answer: 6 groups)
- How many groups of 6 coins can you make? (Answer: 2 groups)
Now, draw a factor rainbow for 12. Start with 1 and 12 at the top, then 2 and 6, then 3 and 4 at the bottom. You're finding all the ways to make 12!
Common mistakes
- Mistake: Thinking that 12 is a factor of 3 because 12 ÷ 3 = 4
- Why it happens: Confusing the relationship between multiplication and division
- Fix: Remember that 3 is a factor of 12 because 3 × 4 = 12
- Mistake: Forgetting that 1 and the number itself are always factors
- Why it happens: Rushing through the process
- Fix: Always start with 1 × that number
- Mistake: Listing multiples in the wrong order
- Why it happens: Not thinking about skip counting
- Fix: Count by the number you're multiplying (e.g., count by 3s for multiples of 3)
Parent and carer guide
Help your child understand multiples and factors by:
- Using real-life examples like sharing cookies or arranging toys in rows
- Practising with small numbers first, then moving to larger ones
- Creating factor rainbows together with different numbers
- Playing games that involve grouping, like "How many groups of 4 can we make?"
- Encouraging your child to explain their thinking to you
Remember, this is a new concept, so be patient. It's normal for children to mix up multiples and factors at first. Keep practicing with fun examples and celebrate small wins!
Practice questions
- What are the first 5 multiples of 6?
- What are all the factors of 18?
- Is 7 a factor of 21? Why or why not?
- Is 15 a multiple of 3? Why or why not?
- Draw a factor rainbow for 20.
Answers:
- Multiples of 6: 6, 12, 18, 24, 30
- Factors of 18: 1, 2, 3, 6, 9, 18
- Yes, because 7 × 3 = 21
- Yes, because 3 × 5 = 15
- Factor rainbow for 20: 1, 2, 4, 5, 10, 20
FAQ
What is the difference between multiples and factors?
Multiples are the results of multiplying a number by integers. Factors are the numbers that multiply together to make a product.
Can a number be both a multiple and a factor?
Yes! For example, 6 is a multiple of 3 (3 × 2 = 6) and 6 is a factor of 12 (6 × 2 = 12).
How do I know if a number is a multiple of another number?
If you can divide the first number by the second number and get a whole number, then the first number is a multiple of the second.
What is a factor rainbow?
A factor rainbow is a visual way to show all the factor pairs of a number by connecting them in a rainbow shape.
Why do we learn about multiples and factors?
They help us solve problems, understand patterns, and make sharing and grouping easier in everyday life.