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ArithmeticThis topic is relevant for:
Here we will learn about factors, including recognising factors, commutativity, how to systematically find all factor pairs of a number, and solving problems using factors.
There are also factors worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβre still stuck.
Factors are numbers that will divide into an integer (a whole number) with no remainder.
Factors are always integers and can sometimes be called divisors. Every integer has at least factors.
If an integer has only two factors, it is a prime number.
For example, the factors of are and
Step-by-step guide: Prime numbers
If an integer has an odd number of factors, it is a square number.
For example, the factors of are and This is because one factor is repeated.
Step-by-step guide: Square numbers
A factor pair consists of two numbers that are factors of another number and make that number when they are multiplied together.
For example, the factor pairs of are,
To find all of the factors of any integer, we write out all of the factor pairs in order.
Factor pairs have a commutative property such that you can switch the order of the calculation and the calculation remains the same:
Prime factors are factors of a number that are also prime numbers.
For example, the prime factors of
Step-by-step guide: Prime factors
Once we know what the factors of a number are, we can use these to solve problems involving probability, substitution, completing tables, and area.
We use factors in other areas of maths including factorising algebraic expressions and quadratics, simplifying and equivalent fractions, calculating the highest common factor for a pair of integers, area calculations, and enlargement.
In order to list all of the factor pairs of a number
Get your free factors worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEGet your free factors worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEFactors is part of our series of lessons to support revision on factors and multiples and factors, multiples and primes. You may find it helpful to start with the main factors and multiples lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
List the factors of
As we have the first factor pair
2Write the next smallest factor of and calculate its factor pair.
As is an even number, is a factor of
so the next factor pair is .
3Repeat until the next factor pair is the same as the previous pair.
So far we have:
and so we have the next factor pair .
and so we have the next factor pair .
which is a decimal so is not a factor of .
The next factor to try is As factors are commutative,
which is the same as the previous factor pair.
We have now found all of the factor pairs:
4Write out the list of factors for
Reading down the first column of factors, and up the second column, the factors of are:
List the factors of
State the pair
As we have the first factor pair
Write the next smallest factor of and calculate its factor pair.
As is an odd number, is not a factor of
However,
and so the next factor pair is
Repeat until the next factor pair is the same as the previous pair.
So far we have:
has a remainder when it is divided by or
The next factor to try is but we already know that is a factor pair and this is the same as
We have now found all of the factor pairs:
Write out the list of factors for
Reading down the first column of factors, and up the second column, the factors of are: and
List the factors of
State the pair
As we have the first factor pair
Write the next smallest factor of and calculate its factor pair.
As is an even number, is a factor of
and so the next factor pair is
Repeat until the next factor pair is the same as the previous pair.
So far we have:
and so the next factor pair is .
and so the next factor pair is .
has a remainder when it is divided by and so is not a factor.
and so the next factor pair is .
We have reached a repeated factor and so we have found all of the factor pairs for
Write out the list of factors for
Reading down the first column of factors, and up the second column, the factors of are: and
Notice that as we have a repeated factor, we have an odd number of factors in the list above. This can help us determine that is a square number.
List the factors of
State the pair
As we have the first factor pair
Write the next smallest factor of and calculate its factor pair.
As is an odd number, is not a factor of
has a remainder when it is divided by and
Repeat until the next factor pair is the same as the previous pair.
The next integer to try is which we have already used
and so the only factor pair for the number is
Write out the list of factors for
The factors of are: and
As has only two factors, and itself, is a prime number.
List the factors of
State the pair
As we have the first factor pair
Write the next smallest factor of and calculate its factor pair.
As is an even number, is a factor of
and so the next factor pair is
Repeat until the next factor pair is the same as the previous pair.
So far we have:
has a remainder when it is divided by and so is not a factor.
and so the next factor pair is
and so the next factor pair is
has a remainder when it is divided by and and so these are not factors of
and so the next factor pair is
has a remainder when it is divided by and so is not a factor.
and so the next factor pair is .
has a remainder when it is divided by and and so these are not factors of
The next integer to try is but this appears in the previous factor pair
We have therefore found all of the factor pairs for
Write out the list of factors for
Reading down the first column of factors, and up the second column, the factors of are: and
The factors of are and By calculating the factors of determine the common factors of and
State the pair
As we want to list the common factors of and we need to find the factors of each of them, and then highlight common factors (the numbers that appear in both lists).
We already have the factors of so we just need to find the factor pairs for
The first factor pair is .
Write the next smallest factor of and calculate its factor pair.
As is an even number, is a factor of
and so the next factor pair is
Repeat until the next factor pair is the same as the previous pair.
8 has a remainder when it is divided by 3.
The next factor to try is 4 and we have already used this in the previous factor pair
We have found all of the factor pairs:
Write out the list of factors for
The factors of are: and
The factors of are: and
The common factors of and are: and
Factors and multiples are easily mixed up. Remember multiples are the multiplication table, whereas factors are the numbers that go into another number without a remainder.
All numbers are a factor of themselves and is a factor of every number.
For example, the factors of are: and so is a factor of itself.
1. List the factors of
and and
The factor pairs of are:
So the factors of are and
2. List the factors of
and
The factor pairs of are:
So the factors of are and
3. List the factors of
and and
The factor pairs of are:
So the factors of are and
4. List the factors of
and
The factor pairs of are:
So the factors of are and
5. List the factors of
and and and
The factor pairs of are:
So the factors of are: and
6. The factors of are and By calculating the factors of determine the common factors of and
and
The factor pairs of are:
So the factors of are: and
The factors of are: and
The common factors of and are: and
1. Here is a list of numbers:
(a) Which numbers are factors of
(b) Which numbers have a common factor of
(c) The product of two numbers is State the two numbers from the list above.
(d) Show that is a factor of
(e) Which number has an odd number of factors?
(5 marks)
(a)
(1)
(b)
(1)
(c)
(1)
(d) and no remainder stated
(1)
(e)
(1)
2. Let and be two single digit prime numbers where:
(a) List the possible values of
(b) Jamal says β is a prime number as it only has factors, and β. Is Jamal correct? Explain your answer.
(5 marks)
(a)
(1)
(1)
(b)
No
(1)
The factors of are and .
(1)
has more than factors and so it is not prime.
(1)
3. A spinner is spun twice. The outcomes are multiplied and written into the sample space diagram below.
(a) Calculate the probability of getting a factor of after spins.
(b) What is the probability of getting a factor of given that the first spin is a
(4 marks)
(a)
Factors of
(1)
(1)
(b)
Factors of
(1)
(1)
4. The Venn diagram below shows the two sets A = Factors of and B = Factors of with some values placed.
(a) Add all of the factors of and all of the factors of to the Venn diagram.
(b) I pick a factor at random. What is the probability that the factor is also a factor of
(5 marks)
(a)
Factors of
(1)
Factors of
(1)
Completed Venn diagram:
(1)
(b)
Factors of
(1)
(1)
You have now learned how to:
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