When it comes to mathematics, many people get confused between the difference between a factor and a multiple. In this article, we will clear up that confusion and help you understand the difference.

A factor is simply a number that is multiplied by another number to produce a result. A multiple is the result of multiplying two numbers together.

As a high school maths teacher for over 10 years this is an important concept to understand for other topics such as fractions and algebra. We hope this article helps you understand these concepts better!

## What is a Factor?

A factor is a whole number that can be divided into a larger number without any remainder.

In other words, factors are the numbers you multiply together to get another number. For example, the factors of 5 are 1 and 5.

The number 5 can be divided exactly by each of these numbers with no remainder.

## What is a Multiple in Maths?

A multiple is the result of multiplying two numbers together.

For example, 10 is a multiple of 2 because 2 x 5 = 10.

Multiples are also sometimes referred to as products. So, you could say that the product of 2 and 5 is 10.

I like to think of multiples as the answers to the times tables.

## Key differences between Factors and Multiples.

- Factors are the divisors of a number whereas multiples are the products of the number. e.g the factors of 10 are 1, 2, 5 & 10 whereas the multiples of 10 are 10, 20, 30, 40, 50, etc
- The factors of a number are finite, whereas the multiples of a number are infinite. e.g There are exactly 4 factors of 10, however the multiples of 10 keep going on and on.
- The factors of a number are less than or equal to the number, whereas the multiples of a number are greater than or equal to the number. See how all the factors of 10 are less than or equal to 10 and the multiples of 10 are greater than or equal to 10.
- The operation we use to find factors is division, whereas the operation we use to find multiples is multiplication.

## How to find Factors

- Draw a factor rainbow to find factor pairs.
- Draw a factor tree to find the prime factors of a number like in this example of the prime factorization of 30.
- Use the divisibility tests like shown in this article to find the factors of 18.

## How to find Multiples

The easiest way to find the first few multiples of a number is to just multiply the number by 1, then 2, then 3 etc.

10 x 1 = 10

10 x 2 = 20

10 x 3 = 30

10 x 4 = 40

10, 20 , 30, 40 are the first 4 multiple of 10.

You can find any multiple of 10 by multiplying 10 by any whole number (integer).

## Factors and Multiples games

Math games are important for a variety of reasons and can help you learn the difference between factors and multiples.

Some math games are visual and hands on which make it easier for visual and kinesthetic learners to understand complex concepts.

This factors and multiples games is great for the classroom for students to play in pairs or for a teacher to play 1 on 1 with a student where you can give them tips and support to find the factors and multiples options when it is their turn.

I play this at school using 100 grids on mini whiteboards with counters.

It is a great opportunity to demonstrate how to find factors by drawing factor rainbows. Give each student a mini whiteboard to draw the factor rainbow.

A similar game that is suitable to play as an individual is this online Factors and Multiples game from nRich Maths where you need to make the longest chain of alternating factors and multiples.

Make it into a competition to see who can make the longest chain and then have a class discussion afterwards about different strategies used which will bring up a rich discussion about prime and composite numbers.

## Related questions

### What is the difference between factors and prime factors?

A factor includes ALL the numbers that divide exactly into a number whereas a prime factor is only the factors that are prime numbers.

### Are ‘factors’ the same as ‘factoring’?

Factors is a noun. Factoring or factorize is the verb.

When you factorize an expression or number you break it up into the product of 2 or more factors.

### What is the difference between a ‘divisor’ and a ‘factor’?

A divisor and factor are the same thing. A larger number can be divided by both a divisor or factor.

## Final Thoughts on the difference between a factor and a multiple

In this article, we’ve explored what factors and multiples are, how they differ, and a few ways you can practice applying them to improve your math skills.

I hope you found this information helpful and that it will help you in your future math endeavors. As always, if you have any questions or feedback, don’t hesitate to reach out to me. Thanks for reading!

## Further factors

Learn more about the factors of other numbers here.