How to find the factors of a number

Key points

are that divide exactly into a number.
Factors can be found by listing them out, using or using
A is a set of two factors. When multiplied together, the pair give a particular . A has one factor pair consisting of one factor multiplied by itself.
A number that has more than two factors is a . This can be expressed as a unique . This is useful when finding the (HCF) and (LCM) of large numbers.
To be able to find factors and write a number as a product of prime factors, having knowledge of powers and indices is useful.

How to find factors and factor pairs using arrays

Draw different rectangular with the right amount of squares. Eg, to find the factors of 15, draw arrays with 15 squares
- The first rectangle will always be 1 x the number you are finding factors for.
- For the next rectangles, consider whether they can have a dimension of 2, 3, 4 etc. Use your understanding of to help.
Each time you draw a rectangle, two are found. These are a
Sometimes one of the rectangles is a square with equal side lengths. This happens when the number is a . The same factor is found twice - therefore only one factor is actually found. Eg, 5 × 5 = 25 and 5 is a factor of 25

Remember: To find all the factor pairs, it is not necessary to draw every rectangle. Eg, a 5 x 2 rectangle and a 2 x 5 rectangle are the same rectangle in a different orientation. Only one of these rectangles is needed to get the factor pair of 2 and 5. Either rectangle can be picked.

Examples

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Use arrays to find the factors and factor pairs of 15
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Draw rectangles containing 15 squares. The number of squares along each side will be a factor of 15
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These rectangles show that 1 x 15 = 15 and 15 x 1 = 15
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These rectangles show that 3 x 5 = 15 and 5 x 3 = 15
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Two unique rectangles can be drawn for 15. The 1 x 15 rectangle is the same as 15 x 1. 3 x 5 is the same as 5 x 3
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15 is a composite number - it has more than two factors. The factors of 15 are 1, 3, 5 and 15. The factor pairs for 15 are the pairs of dimensions for each rectangle. The factor pairs of 15 are 1 and 15, and 3 and 5
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Use arrays to find the factors and factor pairs of 9
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These rectangles show that 1 x 9 = 9 and 9 x 1 = 9. These rectangles give the same factor pair, 1 and 9. Only one of the rectangles is needed - either can be picked.
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This array is a square because the sides are equal in length. The square shows that 3 × 3 = 9. This gives just one factor, 3
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The unique arrays are 1 x 9 and 3 x 3. 9 is a square number and has an odd number of factors - the factors of 9 are 1, 3 and 9. The factor pairs of 9 are 1 and 9, 3 and 3

Question

Use arrays to find all the factors of 24

You will need a pen and paper for this.

How to find prime factors of a number using a factor tree

A is made up of that are numbers greater than 1. A number does not produce a factor tree because one of its factors is 1 and the other is itself so the number would be repeated – the factor tree would not grow.

Write the number at the top of the factor tree.
Draw two branches from the number to split the number into a pair of factors, greater than 1
Write each factor at the end of each branch.
- If a factor is a prime number, the branch does not extend any further and the factor is circled to show this.
- If the factor is not a prime number, it is split into a further pair of factors, greater than 1
This process continues until the branches all end in prime numbers. These are of the number.

Remember: the factor tree of a number can look different, but the numbers at the end of the branches will be the same.

Example

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Find the prime factors of 30 using a factor tree.
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Draw two branches from the number 30 to split it into a pair of factors, greater than 1. The factor pair could be 2 and 15, 3 and 10 or 5 and 6. The factor pair of 1 and 30 is not used because the factor 1 is not greater than 1 and the 30 is repeated.
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In this example the factor pair of 5 and 6 is used. Write each factor (5 and 6) at the end of each branch.
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5 is a prime number so the 5 branch does not extend any further. Circle the 5 to show this.
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6 is not a prime number so the factor tree can grow further. 6 is split into a further pair of factors, greater than 1. 2 x 3 = 6, so write the factors 2 and 3 at the end of each branch.
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2 and 3 are both prime numbers so they can be circled. All the branches now end in prime numbers. The factor tree for 30 is complete. The prime factors are 5, 2 and 3
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The factor tree for 30 can take different forms, depending on which factors are used at the start. The prime factors are always the same. The prime factors of 30 are 2, 3 and 5

Question

Draw a factor tree to find the prime factors of 105

You will need a pen and paper for this.

Write a number as a product of its prime factors

To write a number as a , the number is written as the result of multiplying its together.

Draw a factor tree.
Write the number equal to the factors multiplied together in numerical order.
When a number occurs more than once, this can also be written in form (with powers).

Writing a number as a product of its prime factors can be helpful when finding the (HCF) and (LCM) of large numbers.

Examples

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Use the factor tree of 30 to write 30 as a product of its prime factors.
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The prime factors of 30 are 2, 3 and 5. 30 can be written as a product of its prime factors. The prime factors are written in numerical order. 30 = 2 × 3 × 5
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The prime factors of 24 are 2 and 3
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The prime factors of 24 are 2 and 3. 24 can be written as a product of its prime factors. The prime factors are written in numerical order. 24 = 2 x 2 x 2 x 3
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24 as a product of prime factors can also be written in index form (with powers). This is because 2 occurs more than once. 24 = 2³ × 3
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The prime factors of 750 are 2, 3 and 5
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The prime factors of 750 are 2, 3 and 5. 750 can be written as a product of its prime factors. The prime factors are written in numerical order. 750 = 2 x 3 x 5 x 5 x 5
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750 as a product of prime factors is 2 x 3 x 5 x 5 x 5. This can also be written in index form (with powers) as 5 occurs more than once. 750 = 2 × 3 × 5³

Question

Using the factor tree, write 108 as a product of its prime factors.

A factor tree diagram of numbers branching from one-hundred and eight. One-hundred and eight going to nine and twelve. Nine going to three and three – both circled. Twelve going to a circled two and a six. Six going to two and three – both circled.

Practise finding factors

Practise what you've learned about finding the factors of a number with this quiz. You may need a pen and paper for some of these questions.

Quiz

Real-world maths

Finding factors and factor pairs can be useful in many different situations in real life.

For example, a couple are getting married and are planning their wedding. They have 72 guests coming who all need a seat. The tables can seat up to 12 people.

A wedding venue with tables ready for guests. — Image caption,
A wedding venue ready for guests to arrive.

By using factor pairs, the couple can choose different combinations to suit their friends and family:

12 tables of 6 guests
9 tables of 8 guests
8 tables of 9 guests
6 tables of 12 guests

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