For many students, maths is the one subject in school that they find the most difficult to understand. However, once you put your mind to it, maths does not need to be as daunting as you might think. Once the basics are mastered, everything else begins to fall into place. That is why grasping factors is so important.
There are lots of little tips and tricks that you can use to find the factors of a number. Once you understand factors, this will help you with other areas of maths, such as fractions and algebraic equations. Learning about fractions in a class can be overwhelming, which is why we have created this handy guide with everything you need to know about factors, how to find them and how to use them, along with examples.
If you are struggling with maths or any other subjects in class, consider seeking the help of a tutor who can help you reach your full academic potential. Tutors can cater their lessons to your learning style by using games, worksheets and important resources to help you understand key concepts.
What is a factor?
A factor is a number that divides perfectly into another number. By saying it divides perfectly into another number, we mean that when divided into a larger number, there are absolutely no remainders.
For example, 2 divides perfectly into 4 and there are no additional numbers left over. It is also important to remember that you can either have positive factors or negative factors, but they have to be a whole number. Every single number has at least two factors apart from the number 1 and 0.
Each number is guaranteed to have a factor pair of 1 and the number itself, because both the number 1 and the number itself can always be divided into the number. For example, 7 is a prime number so it can only be divided by 1 and itself, meaning the factors of 7 are 1 and 7.
Although even prime numbers have factors, you can also get prime factors. These are factors that are in themselves prime numbers. For example, 21 is not a prime number, but it does have prime factors. Both 7 and 3 divide into 21, making them factors of 21. In addition to this, they are both prime numbers as only 1 and itself can be divided into 7 and 3, making them prime factors.
Properties of factors
Knowing the properties of a factor of a number can help you identify them when you need to in larger equations in maths and even science where finding common factors is often needed. Below are some of the properties of factors.
One is a factor of every number
Any given number can be divided by one, even the number one can be divided by one, meaning that by default, one is a factor of every number. You will also find that 1 is the smallest factor of every number.
For example:
There are 7 ones in 7
There are 21 ones in 21
There are 18 ones in 18
This goes on for each and every number.
The number itself is a factor of every number
Similar to the number one, every number can be divided by the number itself, meaning that the number itself is always a factor of itself. Once again, you will find that the number itself will always be the biggest factor of that number.
For example:
There is one 7 in 7
There is one 21 in 21
There is one 18 in 18
Every number apart from 0 and 1 have at least 2 factors
As the previous two properties highlight, every number apart from 0 and 1 has at least 2 factors as they can always be divided by both 1 and themselves. If a number only has these two factors, it means that it is a prime number.
Factors can be positive or negative
Although this can be confusing to some students, the concept of factors being either positive or negative is actually extremely simple. When you think about it, two negative numbers will always make a positive, so it is really no different than having two positive numbers and the answer remains the same.
For example:
Let us take the number 7.
The positive prime numbers of 7 are 7 and 1.
The negative prime numbers of 7 are -7 and -1.
Because, 7 x 1 = 7
AND -7 x -1 = 7 too
There are a finite amount of factors
No number can have an unlimited amount of factors because the highest number a factor can be is the number equal to itself. Therefore, you cannot go on and on with factors. You will always be able to find all the factors of a number as it will end at some point.
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How to find the factors of a number
Now that you know exactly what a factor is and what properties to look for, the next step is to understand how exactly to find factors.
Find factors through division
The first method that you can use to find the factors of any given number is division. All you need to do is take a number and note down all the numbers that are less than that number. You then divide each number into your original number and if it divides perfectly into your original number without any remaining numbers, it is a factor of that number.
For example:
If you are trying to find all the prime numbers of 6, you need to take all of the numbers under or equal to 6. So, you gather the numbers 1,2,3,4,5 and 6. You then work through these numbers and see which ones divide perfectly into 6 without any remainders:
6 1 = 6
6 2 = 3
6 3 = 2
6 4 = 1 remainder 2
6 5 = 1 remainder 1
6 6 = 1
The factors of 6 are 1, 2, 3 and 6 as these are the numbers that can divide into 6 without having any remaining numbers left over.
Find factors through multiplication
To find factors through multiplication, you have to go through trial and error as you take your given number and try out all the possible pairs of numbers that multiply by each other to make that number. Both of the numbers in that sum will be prime numbers of the original integer.
For example:
Let us take the example of 6 again. If you are trying to find all the factors of 6 through the multiplication method, you need to see what number multiplied by each other equal 6:
1 x 6 = 6
2 x 3 = 6
6 x 1 = 6
3 x 2 = 6
Whilst you have 4 different sums that give you the answer 6, you see that some of the numbers are duplicated, you do not need to include these duplications in your answer and only need one of each number. Therefore, from the above equations, we can see that the prime numbers of 6 are 1, 2, 3 and 6.
Remember that with either method of finding fractions, your answers can be both positive or negative so the answers that you get that are positive can be the same negative numbers. Therefore, for the above examples, just as 1, 2, 3 and 6 are factors, so are -1, -2, -3 and -6.
Prime factorization
Most students will also need to know how to perform prime factorization, which is a little more complicated. Prime factorization is an exploration into which prime numbers multiply together to make the original number.
In order to find the solution, you need to take your original number, divide it by the smallest prime number, see if it divides perfectly, then see if your answer is a prime number. If it is not, you need to keep going until your answer is a prime number.
For example:
Find the prime factorization of 12.
First, you need to divide 12 by the smallest prime number, which is 2.
12 2 = 6
6 is not a prime number so you need to repeat the steps with the 6.
6 2 = 3
3 is a prime number, so you have reached the end. The sum that you create from this process is:
12= 2 x 2 x 3, as you used two 2s and a 3.
This can also be written as 12= 22 x 3
Examples of factors
To help you get some practice in before class, we have created this list of examples to show you exactly how you can find the factors of different numbers.
What are the prime numbers of 14?
If you are going to use the multiplication method, you need to see which numbers multiplied together will give you 14, as follows:
1 x 14 = 14
2 x 7 = 14
These are the only numbers that multiply into 14, meaning the factors of 14 are 1, 2, 7, and 14. You can also have the negative numbers if these positive ones, meaning -1, -2, -7 and -14 are also factors of 14. 14 is not a prime number but 7, 2 and 1 are prime factors.
What are the prime numbers of 10?
If you are going to use the division method, you need to gather all the numbers under 10 and see which divide into the original number.
10 1 = 10
10 2 = 5
10 3 = 3 remainder 1
10 4 = 2 remainder 2
10 5 = 2
10 6 = 1 remainder 4
10 7 = 1 remainder 3
10 8 = 1 remainder 2
10 9 = 1 remainder 1
10 10 = 1
You therefore see that the only numbers that divide perfectly into 10 without any remaining number are 1, 2, 5, and 10. Therefore, the factors of 10 are 1, 2, and 10 and -1, -2, -5 and -10.