Multiples of 9

Writing and Forming Algebraic Expressions

how to find a missing number in a sequence

A number sequence is a list of numbers that follow a pattern.

An example of a number sequence is 5, 9, 13, 17.

To find the difference between each number, subtract one number from the number that comes after it.

9 – 5 = 4 and so, the common difference is 4.

Each number in a sequence is called a term.

To find the next term in the sequence, add 4 to 17 to make 21.

how to find a missing term in a sequence

A sequence can be formed by adding the same number each time.

Subtract one number from the number that comes after it to find this common difference.

example of finding a missing term in a sequence

To find the difference between each number, we subtract one term from the term that is immediately after it.

32 – 22 = 10 and so, this sequence is going up by 10 each time.

We can add 10 to 32 to find the missing number of 42.

Alternatively, we can subtract 10 from 52 to find 42.

Supporting Lessons

How to Find the Next Number in a Number Sequence

How to Find a Missing Number in a Linear Number Sequence

How to Find the Common Difference in a Number Sequence

Number Sequences

What is a Number Sequence?

A number sequence is a pattern of numbers formed using a rule. For example, 1, 4, 7, 10 is a number sequence that is formed by starting at 1 and adding 3 each time. We can add 3 more to find the next number in the sequence, which is 13.

Each number in a sequence is called a term.

A number sequence is made up of different terms. The example below shows a sequence made up of 5 terms.

example of an arithmetic sequence going up in fours

An arithmetic sequence is a number pattern made by adding or subtracting the same amount to get from one number to the next. For example, in the sequence of 1, 4, 7, 10, 13, we add 3 to get from one number to the next.

how to find the next term in a sequence

To write a number sequence, write each term separated by a commma.

Some examples of basic number sequences are the times tables. The sequence formed by the 2 times table begins 2, 4, 6, 8, 10.

the 2 times table sequence

The two times table goes up in twos each time. We can say that we add 2 to get from one term to the next.

How to Find the Next Number in a Sequence

To find the next number in a sequence, first find the common difference by subtracting one term from the term that comes immediately after it. Add this common difference on to the last term in the sequence to find the next number in the sequence.

For example, in the sequence, 3, 5, 7, 9, the difference from one term to the next is 2. To find the next number in the sequence, add 2 to 9 to make 11.

how to find the next number in the sequence example

We can find the common difference by subtracting any two numbers that are next to each other. 5 – 3 = 2 and 9 – 7 = 2. The common difference is 2.

We add 2 to get from each number to the next.

how to find the next term in a sequence

Here is an example of finding the next number in a sequence.

example of finding the next term in an arithmetic sequence

The first step is to find the common difference.

We subtract one term from the term that comes after it.

9 – 5 = 4 and so, the common difference is 4.

example of finding the next number in a sequence

We add 4 on to the final number in the sequence to find the next term.

17 + 4 = 21 and so, the missing term in this sequence is 21.

We can continue to add 4 on to continue the sequence. We would get the following sequence: 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 59, 63 and so on.

How to Find a Missing Term in an Arithmetic Sequence

To find a missing term in an arithmetic sequence, first identify the common difference by subtracting any term from the term that comes immediately after it. Count up or down by this amount from term to term until you find the missing value needed.

An arithmetic sequence is simply a list of numbers that are formed by adding on the same amount each time to get from one number to the next.

Here is the sequence 3, _, 7, 9, 11. The second term is missing.

finding a missing second term in a sequence

To find the common difference, we subtract any term from the term that comes after it.

11 – 9 = 2 and 9 – 7 = 2. We can see that the common difference is 2.

finding a missing term within a sequence example

We can add 2 to 3 to find the missing term of 5.

Alternatively, we can subtract 2 from 7 to find the missing term of 5.

Here is an example of finding 2 missing terms in a sequence.

We have the sequence _, 10, _, 24, 31.

working out missing terms in a sequence

We first need to find the common difference between two terms. We need to look at two consecutive terms, which are two numbers in the sequence that are next to each other.

how to find missing terms in a sequence example

31 – 24 = 7 and so, the common difference of this sequence is 7.

We can count down in sevens to find the missing terms.

24 – 7 = 17 and 10 – 7 = 3. 17 and 3 are the missing terms in the sequence.

How to Find the Common Difference of a Sequence

The common difference is simply the difference between each term in a sequence. To find the common difference, subtract one term from the term that comes after it. If there are no two consecutive terms, then find the difference between any two terms and divide this by the number of jumps between these terms.

For example, here is the sequence 10, _, _, _, 50.

how to find the common difference in a sequence

The difference between 50 and 10 is 40. We divide this difference by the number of jumps between the two terms.

finding the common difference between two terms in a sequence

There are 4 jumps between the terms of 10 and 50. Therefore we divide the difference of 40 by 4 to share it evenly over these 4 jumps.

40 ÷ 4 = 10 and so the common difference is 10.

We can add on in tens to find the missing terms.

The completed sequence is 10, 20, 30, 40, 50.

Here is another example of working out the common difference in an arithmetic sequence.

We have 2, _, 10, _, _.

The two known terms are 10 and 2.

calculating the common difference in an arithmetic sequence

10 – 2 = 8 and so, the difference between the terms is 8. However this difference needs to be shared over two jumps.

8 ÷ 2 = 4 and so, each jump is worth an addition of 4.

We add on in fours to get from one term to the next.

example of using the common difference to find missing terms

The completed sequence is 2, 6, 10, 14, 18.

Here is another example of finding the missing terms in a sequence when we only have 2 known terms.

We have _, 7, _, _, 28.

The difference between 28 and 7 is 21.

finding the missing terms in a sequence

This difference of 21 is shared equally over 3 jumps.

21 ÷ 3 = 7. The common difference is 7.

We can add on in sevens to find the missing terms.

missing numbers in a sequence finding the term to term difference

The sequence is 0, 7, 14, 21, 28.

How to Find the nth Term of a Sequence

To find the nth term of a sequence, follow these steps:

Find the difference from one term to the next in the sequence.

Multiply this difference by ‘n’ and write this down.

Decide what number must be added to or subtracted from the difference found in step 1 in order to make the first term of the sequence.

Add this number to the end of the expression written in step 2.

The nth term is a formula that is used to create a sequence. Simply substitute values in place of n to find the relevent term. To find the first term, substitute n = 1. To find the second term, substitute n = 2.

Here is the 2n sequence.

the two times table nth term

The 2n sequence begins with 2, 4, 6, 8 and so on.

The 2n sequence is the two times table. The difference between each number is 2.

Here is an example of finding the nth term of the sequence 2n + 3.

The sequence is 5, 7, 9, 11.

The first step is to find the common difference between each term. This term is going up in twos.

The second step is to write this difference multiplied by n. We have 2n.

how to find the nth term of a sequence example of 2n + 3

The third step is to decide what number should be added to this difference to make the first term.

The difference is 2 and the first term is 5. We need to add 3 to 2 to make 5.

The final step is to write this on the end of the expression. 2n becomes 2n + 3.

We can see that the 2n + 3 sequence goes up in twos like the 2n sequence, however all numbers are 3 larger.

Here is another example of finding the nth term of a sequence.

We have 3, 8 13, 18.

The difference between each term is 5. So we start with the 5n sequence.

We need to subtract 2 from 5 to make the first term of 3 and so the sequence is 5n – 2.

finding the nth term of the sequence 5n
- 2

The 5n sequence is the 5 times table and so, it goes up in fives each time.

The 5n – 2 sequence also goes up in fives each time but the numbers are all 2 smaller than the 5 times table.

Now try our lesson on Writing and Forming Algebraic Expressions where we introduce algebraic expressions.

example of an algebraic expression

Multiples of 9

multiples of 9 list

Multiples of 9 are numbers that can be divided exactly by 9.

The first few multiples of 9 are the numbers in the 9 times table.

For example 3 × 9 = 27 and so, the third multiple of nine is 27.

For example 100 × 9 = 900 and so, the hundredth multiple of 9 is 900.

The first multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99 and 108.

A multiple of 9 is any number that can be divided by 9 exactly, leaving no remainder.

Multiples of 9 Flashcards

Click on the multiples of 9 flashcards below to memorise the 9 times table:

Supporting Lessons

Multiples of 9: Interactive Questions

Multiples of 9 Worksheets and Answers

Finding Factors of a Number

Multiples of 9

What are the Multiples of 9?

The multiples of 9 are any numbers that can be divided exactly by 9 to leave no remainder. The first ten multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81 and 90. We can continue to find more multiples of 9 by adding on 9 each time. There are an infinite number of multiples of 9.

Here is the 9 times table, which makes up the first 12 multiples of 9:

first multiples of 9

The multiples of 9 all have a difference of 9 between them. Add 9 from one multiple of 9 to get to the next multiple of 9.

multiples of 9 list

Patterns in the Multiples of 9

The multiples of 9 all have digits that add up to another multiple of 9. There is also a diagonal pattern created by shading the multiples of 9 on a number grid.

Here are the multiples of 9 shown shaded on a number grid.

diagonal pattern in the multiples of 9

We can see that there is a clear diagonal pattern shown in the multiples of 9. This pattern can help us figure out the next multiple of 9.

When teaching the multiples of 9, it can be helpful to show a number grid, where the diagonal pattern is shown.

We can see further patterns in the multiples of 9 by reading their digits.

The digit at the end of the number decreases by 1 as we move from one multiple of 9 to the next. For example, 9 ends in 9, 18 ends in 8 and 27 ends in 7.

patterns in the multiples of 9

Another pattern is that the number in the tens column decreases by 1 as we move from one multiple of 9 to the next. For example, the tens digit in 9 is 0, in 18 it is 1, in 27 it is 2 and so on.

An interesting pattern in the multiples of 9 is that the sum of the digits add to a multiple of 9. The first 10 multiples of 9 have digits that always sum to make 9, which can make them easier to memorise.

For example, when looking at the numbers of 9, 18, 27 and 36, we have 9 = 9, 1 + 8 = 9, 2 + 7 = 9 and 3 + 6 = 9. This pattern continues all the way up to 90, which has 9 + 0 = 9.

With larger multiples of 9, there is a pattern of the digits adding to any multiple of 9.

For example, with 12 × 9 = 108, we can see that the sum of the digits adds to 9. 1 + 0 + 8 = 9.

pattern in the digits of the multiples of 9 example of 108

The sum of the digits add to 9 or any other multiple of 9.

rule to decide if a number is a multiple of 9

Here is the example of 21 × 9 = 189.

showing that 189 is a multiple of 9

Here we can see that 1 + 8 + 9 = 18. Again the sum of the digits of this multiple of 9 make another multiple of 9. 18 is 2 × 9.

how to test if a number is a multiple of 9 example of 189

How to Identify Multiples of 9

All multiples of 9 must have digits that sum to make a smaller multiple of 9. For example, 5976 is a multiple of 9 because 5 + 9 + 7 + 6 = 27 and 27 is 3 × 9. 3299 is not a multiple of 9 because 3 + 2 + 9 + 9 = 23, which is not a multiple of 9.

To decide if a number is a multiple of 9, follow these steps:

Add the digits of the number.
If the result is a multiple of 9, then the original number is a multiple of 9.
If the result is not a multiple of 9, then the original number is not a multiple of 9.
If you are not sure if the result is a multiple of 9, add the digits of this number and use the above steps to decide.

For example, here is 5976.

The first step is to add the digits.

5 + 9 + 7 + 6 = 27.

27 is a multiple of 9 because 3 × 9 = 27.

rule to identify multiples of 9

If you were unsure if 27 was a multiple of 9, then you can add the digits of this number. 2 + 7 = 9 and so, 27 is a multiple of 9.

Because the sum of the digits of 5976 is a multiple of 9, we know that 5976 is also a multiple of 9.

rule to identify if a number is a multiple of 9

Here is an example of a number that is not a multiple of 9. We will use the rule to show that 3299 is not a multiple of 9.

The first step is to add the digits.

3 + 2 + 9 + 9 = 23.

23 is not a multiple of 9 and so, 3299 is not a multiple of 9.

We know that 23 is not a multiple of 9 because 2 + 3 = 5, which is not a number in the 9 times table.

example of a number that is not a multiple of 9

The rule for identifying multiples of 9 works for all numbers, no matter how large.

Here is another example, testing if the number 7635 is a multiple of 9 or not.

testing if a number is a multiple of 9. example of a number that is not a multiple of 9.

7 + 6 + 3 + 5 = 21. 21 is not a multiple of 9 and so 7635 is not a multiple of 9. We know that 21 is not a multiple of 9 because 2 + 1 = 3, which is not in the 9 times table.

Is Zero a Multiple of 9?

Zero is a multiple of 9 because it can be divided exactly by 9 to leave no remainder. 0 ÷ 9 = 0. We can also see that zero is a multiple of 9 because 0 × 9 = 0. Zero is a multiple of every number.

What are the Multiples of 9 up to 100?

There are 11 multiples of 9 that are smaller than 100. They are:

9, 18, 27, 36, 45, 54, 63, 72, 81, 90 and 99.

list of the multiples of 9 to 100

Multiples of 9 List

Here is a list of multiples of 9:

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 279, 288, 297, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405, 414, 423, 432, 441, 450, 459, 468, 477, 486, 495, 504, 513, 522, 531, 540, 549, 558, 567, 576, 585, 594, 603, 612, 621, 630, 639, 648, 657, 666, 675, 684, 693, 702, 711, 720, 729, 738, 747, 756, 765, 774, 783, 792, 801, 810, 819, 828, 837, 846, 855, 864, 873, 882, 891, 900, 909, 918, 927, 936, 945, 954, 963, 972, 981, 990 and 999.

There are 111 multiples of 9 up to 1000.

Now try our lesson on Finding Factors of a Number where we learn how to find factors.

factors

Multiples of 9: Interactive Questions

Multiples of 9: Interactive Questions Return to video lesson on Multiples of 9

Units of Length

metric units of length

A millimetre is the smallest measurement on a ruler. A millimetre is about as thick as a staple or the tip of a pencil.

Millimetres are usually used to measure objects that can be held in one hand. They can also be used to measure larger objects that need greater accuracy, such as the size of a phone screen.
There are 10 millimetres in a centimetre.

Centimetres are usually used to measure objects that are smaller than a ruler. They can also be used to measure larger objects with greater accuracy, such as a person’s height.

There are 100 centimetres in a metre.

Metres are usually used to measure objects that are longer than the width of a doorway but they can be used to measure smaller objects too.

There are 1000 metres in a kilometre.

Kilometres are usually used to measure distances between places such as towns, cities and countries as well as the length of roads.

The most common metric units of length are kilometres (km), metres (m), centimetres (cm) and millimetres (mm).

There are 10 mm in 1 cm, 100 cm in 1 m and 1000 m in 1 km.

The common imperial units of length are inches (in), feet (ft), yards (yd) and miles (mi).

These units of length are most commonly used in the USA.
There are 12 inches in a foot.

There are 3 feet in a yard

There are 1760 yards in a mile.

an example of converting 8 cm to 80 mm by multiplying by 10

There are 10 millimetres in every centimetre.

8 cm can be written in millimetres by multiplying it by 10.

8 x 10 = 80 and so, 8 cm = 80 mm.

Supporting Lessons

Choosing Appropriate Units of Length Worksheets and Answers

Measuring Centimetres Using a Ruler

Units of Length

Metric Units of Length

From smallest to largest, the metric units of length are millimetres (mm), centimetres (cm), metres (m) and kilometres (km). There are 10 mm in 1 cm, 100 cm in 1 m and 1000 m in 1 km. The metric units of length are used in all countries except the USA, Liberia and Myanmar.

Here is a table showing the most common metric units of length and how to convert between them.

To convert between metric units of length, use the following rules:

To convert cm to mm, multiply by 10.

To convert m to cm, multiply by 100.

To convert km to m, multiply by 1000.

To convert mm to cm, divide by 10.

To convert cm to m, divide by 100.

To convert m to km, divide by 1000.

Here are some examples of converting between metric units:

Original Metric Unit	Conversion	Equivalent Metric Unit
20 mm	÷ 10	2 cm
36 mm	÷ 10	3.6 cm
200 cm	÷ 100	2 m
518 cm	÷ 100	5.18 m
5000 m	÷ 1000	5 km
7200 m	÷ 1000	7.2 km
8 km	× 1000	8000 m
6.3 km	× 1000	6300 m
5 m	× 100	500 cm
3.4 m	× 100	340 cm
9 cm	× 10	90 mm
6.5 cm	× 10	65 cm
3 km	× 100 000	300 000 cm
2.5 m	× 1000	2500 mm

Imperial Units of Length

From smallest to largest, the most common imperial units of length are inches (in), yards (yd), feet (ft) and miles (mi). There are 12 inches in a foot, 3 feet in a yard and 1760 yards in a mile. Imperial units of length are used in the USA, Liberia and Myanmar.

Here is a table showing the most common imperial units of length and how to convert between them.

To convert between metric units of length, use the following rules:

To convert feet to inches, multiply by 12.

To convert yards to feet, multiply by 3.

To convert miles to yards, multiply by 1760.

To convert inches to feet, divide by 12.

To convert feet to yards, divide by 3.

To convert yards to miles, divide by 1760.

What is the Difference Between Metric and Imperial Units?

Metric units use base 10 conversions, whereas imperial units do not. This means that it is easier to convert between metric units by multiplying or dividing by 10, 100 or 1000.

Metric units have a consistent conversion ratio across different measurements whereas imperial units do not. This makes it easier to remember and work with metric units. For example, to convert from kilometres to metres, we multiply by 1000 and to convert from kilograms to grams, we also multiply by 1000. We do not have similar patterns with imperial units such as inches to feet or ounces to pounds, which can make it more challenging to work with these units.

Metric units are used consistently in most countries throughout the world. Only America, Libera and Myanmar officially use imperial units. These countries still use these traditional imperial units because they have been working with imperial units for many years and it would be costly to convert everything into metric units.

Although similar in size, a metre is not the same length as a yard. 1 m is equal to 1.09 yards and so, 1 metre is longer than 1 yard by approximately 3 inches or 8 centimetres.

A centimetre is smaller than an inch. A centimetre is less than half the size of an inch. 1 inch is approximately equal to 2.5 centimetres.

Units of Length List

The most common units for length are shown in the list below with examples:

Metric Units of Length	Example	Conversion
1 millimetre (mm)	Thickness of a staple	10 mm = 1 cm
1 centimetre (cm)	The width of a little finger	100 cm = 1 m
1 metre (m)	The width of a doorway	1000 m = 1 km
1 kilometre (km)	Roads are measured in kilometres	1000 m = 1 km
Imperial Units of Length	Example	Conversion
1 inch	The size of television screens	12 inches = 1 foot
1 foot	People’s height	3 feet = 1 yard
1 yard	The size of rooms or buildings	1760 yards = 1 mile
1 mile	The distance between cities	1760 yards = 1 mile

Other less commonly used units of length are nanometres and micrometres. These units of length are used to measure incredibly small distances as seen under a microscope.

In imperial units, 1 thou is one thousandth of 1 inch. 1000 thous make up 1 inch.

3 miles is called 1 league.

How to Choose an Appropriate Unit of Length

From smallest to largest, the common units of length are millimetres, centimetres, metres and kilometres.

Millimetres are typically used to measure very small objects, such as objects you can hold in your closed hand. Centimetres are typically used to measure objects that are shorter than the width of a door. Metres are typically used to measure objects that are longer than the width of a door. Kilometres are used to measure the distances between towns, cities or countries.

Here is an example of measuring a seed in millimetres. The seed is 8 mm long and it is less than 1 cm.

measuring a seed in millimeters

We know that millimeters would be an appropriate choice of unit to measure this seed because the seed is not even 1 centimetre long.

It is always possible to use a smaller unit of measurement than is typically used. This may be done to measure something to a greater degree of accuracy.

For example, a phone screen is large enough to be measured in centimetres but its size might be given in millimetres for a greater level of accuracy.

Here is an example of measuring a pencil in centimetres. The pencil is 12 cm long.

measuring a pencil in centimetres

If an object can be measured well with a ruler, centimetres are a good choice of unit.

It is easier to say 12 cm than 120 mm. The next unit up after centimetres is metres and this pencil is not even 1 metre long.

Metres are a typical unit of length chosen to measure the size of buildings. Below is an example of measuring the size of a house in metres.

measuring a house in metres

Distances between towns, cities and countries are always measured in kilometres.

For example, the distance between the two cities of London and Manchester is approximately 300 km.

measuring distances in kilometers

Here are some examples of selecting the most appropriate unit of measurement:

Object	Most appropriate unit of measurement	Actual measurement
The length of a grain of rice	Millimetres	7 mm
A blueberry	Millimetres	12 mm
An eraser	Millimetres or centimetres	45 mm or 4.5 cm
An apple	Centimetres	10 cm
An A4 page	Centimetres	29.7 cm
A man	Centimetres or metres	180 cm or 1.8 m
A bus	Metres	12 m
The height of a skyscraper	Metres	150 m
The distance between the UK and the USA	Kilometres	7000 km

Examples of Objects 1 mm Long

A millimetre is the smallest measurement that can be made using a ruler.

The size of a millimetre is shown below.

the size of 1 millimeter

Some examples of objects 1 mm long are:

The width of a staple

The width of a sharp pencil tip

The thickness of a piece of card

The thickness of a bank card

The width of a poppy seed

Examples of Objects 1 cm Long

A centimetre is about the width of a little finger. It is the distance between the large numbers on a ruler.

how long is 1 cm?

Some examples of objects 1 cm long are:

The width of your little finger

The length of a staple

The width of a pea

The thickness of notepad

The length of a coffee bean

Examples of Objects 1 m Long

Some examples of objects 1 m long are:

The length of a baseball bat

The length of a staple

The height of a door handle from the ground

The height of a person’s hips from the ground

The lengths of a guitar

The largest footstep you can make

A metre is about as tall as a small child. However a fully grown adult may be just below 2 metres tall.

how big is 1 meter?

Examples of Things that are 1 km Long

Some examples of things that are 1 km long are:

The Golden Gate Bridge

The distance walked in 15 minutes

The world’s tallest skyscrapers

Now try our lesson on Measuring Centimetres Using a Ruler where we learn how to measure objects correctly using a ruler.

measuring centimetres using a ruler

Multiples of 5: Interactive Questions

Multiples of 5: Interactive Questions Return to video lesson on Multiples of 5

Prime Factor Trees

the prime factorization of 24 using a factor tree

A factor tree is a method used to write a number as the product of its prime factors.

This means finding the prime numbers that multiply together to make that number.

A factor is a number that divides exactly into another number

A prime number is a number that has only 2 factors: 1 and itself.

For each number in the tree, find a pair of whole numbers that multiply to make that number.

Every time we reach a prime number, we stop and then circle that number.

24 = 12 x 2.

2 is prime because it can only be made by 1 x 2. We circle it.

We never choose 1 as a factor in our factor trees because we would end up with the same number again and 1 is not primema.

12 can then be written as 2 x 6.

Again, 2 is prime, so we circle it.

6 can be written as 2 x 3.

Both 2 and 3 are prime, so we circle them.

We write 24 as the circled prime numbers multiplied togther.

The prime factorisation of 24 is 24 = 2 x 2 x 2 x 3.

We can write this prime factorisation in index form as 24 = 2³ x 3.

A prime factor tree is a diagram used to find the prime numbers that multiply to make the original number.

prime factorization of 36

36 can be written as 9 x 4.

9 can be written as 3 x 3.

3 is prime because it cannot be divided to make a smaller whole number. We circle the threes.

4 can be written as 2 x 2.

2 is prime because it cannot be divided to make a smaller whole number. We circle the twos.

The prime factorisation of 36 is 36 = 2 x 2 x 3 x 3.

We can write this in index form as 36 = 2² x 3².

It does not matter which numbers were chosen in each stage of the factor tree, the final prime factorisation will always be the same.

Supporting Lessons

Prime Factor Trees Worksheets and Answers

Prime Factor Trees

What are Prime Factor Trees

Prime factor trees are diagrams that show the prime factorisation of a number. The number is broken down into two of its factors. These factors are then broken down into their factors until only prime numbers remain.

A factor is a number that divides exactly into another number. Two factors can multiply to make another number.

A prime number is a number that has only two factors, which are 1 and the number itself. Put simply, a prime number cannot be divided by a whole number to make a smaller whole number.

definitions of prime numbers and factors

Here is an example of a prime factor tree for the number 36.

A factor tree is a diagram which helps us to find the prime numbers that can multiply together to make the original number.

prime factorization of 36 using a factor tree

We first think of two numbers that multiply to make 36.

36 = 9 × 4 and so, we draw two lines coming out of 36 with 9 and 4 at the ends of them. We say that 9 and 4 are factors of 36.

We can continue to split these numbers into smaller factors.

9 can be written as 3 × 3 and 4 can be written as 2 × 2.

Now we circle the threes and the twos because they are prime. We know this because the only way to make 3 by multiplying whole numbers is 1 × 3 and the only way to make 2 by multiplying whole numbers is 1 × 2.

We write 36 as these circled prime numbers multiplied together.

36 = 2 × 2 × 3 × 3.

We can check this by multiplying the numbers. 2 × 2 = 4 and 3 × 3 = 9. 4 × 9 = 36.

Common Mistakes

A common mistake when drawing factor trees is to write a prime number as 1 × itself. We do not include 1 in the prime factor tree because 1 is technically not a prime number. It also does not help us because we just get the same number appearing again.

Another common mistake is to find two numbers that add to make the number above it instead of numbers that multiply to make the number above it.

How to do Prime Factor Trees

To do prime factor trees, follow these steps:

Draw two diagonal lines below the number.

At the ends of the two lines, write two numbers that multiply to make the number above the lines.

Repeat steps 1 and 2 for each new number written unless it is a prime number.

If the number is a prime number, circle it.

When all numbers at the base of the tree have been circled, the method is complete.

Write the original number as the product of the circled prime numbers.

For example, here is a prime factor tree for the number 24.

The first step is to draw two diagonal lines below the number.

The second step is to write two numbers that multiply to make 24. 12 × 2 = 24 and so, we write 12 and 2 at the end of each line.

how to do prime factor trees example of 24

We now repeat these steps for each factor as long as it is not a prime number.

2 is a prime number because it cannot be broken down into any smaller whole numbers. We circle it.

We draw lines below 12 and write it as 6 × 2. Again, 2 is prime so we circle it.

6 can be written as 2 × 3. Both 2 and 3 are prime and so we circle them both.

We write the original number as the product of the circled prime numbers.

24 = 2 × 2 × 2 × 3.

It does not matter which numbers are chosen at each stage of the factor tree. The final prime factorisation will always be the same.

For example, we could write 24 as 4 × 6. We can then write 4 as 2 × 2 and 6 as 2 × 3.

We still arrive at 24 = 2 × 2 × 2 × 3.

How to Write the Prime Factorisation of a Number

To write the prime factorisation of a number, follow these steps:

Write the number as the product of any two of its factors.

If the factors are prime, circle them.

If a factor is not prime, write it as the product of any two of its factors.

Repeat steps 2 and 3 until all new factors written are prime and there are no more factors to find.

The prime factorisation of the number is written as all of the circled prime numbers multiplied together.

For example, here is the prime factorisation of 48.

the prime factorisation of 48

We write 48 as 6 × 8.

We write 6 as 3 × 2 and we write 8 as 4 × 2. 2 and 3 are prime so we can circle these numbers.

We can write 4 as 2 × 2, which again can be circled because they are prime.

To write the prime factorisation of a number, write the prime numbers multiplied together.

48 = 2 × 2 × 2 × 2 × 3.

We can write numbers that have been factorised in index form.

2 × 2 × 2 × 2 = 2⁴ because 4 twos are multiplied together.

We can write 48 = 2⁴ × 3.

It does not matter which numbers are chosen at each stage of the prime factor tree, the prime factorisation of a number is always the same.

Prime numbers are the building blocks of all other numbers and the prime factorisation of a number is unique to that number.

Now try our lesson on What are Square Numbers where we learn what square numbers are.

What are Square Numbers

4 squared

Multiples of 12

multiples of 12 poster

The multiples of 12 are numbers that can be divided exactly by 12, leaving no remainder.

The first few multiples of 12 are the numbers in the 12 times table.

For example 2 × 12 = 24 and so, the second multiple of twelve is 24.

For example 100 × 12 = 1200 and so, the hundredth multiple of 12 is 1200.

A multiple of 12 is any number that can be divided by 12 exactly, leaving no remainder.

Multiples of 12 Flashcards

Click on the multiples of 12 flashcards below to memorise the 12 times table:

Supporting Lessons

Multiples of 12: Interactive Questions

Multiples of 12 Worksheets and Answers

Multiples of 12

What are Multiples of 12

Multiples of 12 are numbers that can be divided exactly by 12 with no remainder. Multiples of 12 are the numbers formed when any whole number is multiplied by 12. For example, the third multiple of 12 is 36 because 12 × 3 = 36.

The first multiples of 12 are found in the twelve times tables. The first multiples of 12 are shown in the following poster.

multiples of 12

The first multiples of 12 are:

1 × 12 = 12
2 × 12 = 24
3 × 12 = 36
4 × 12 = 48
5 × 12 = 60
6 × 12 = 72
7 × 12 = 84
8 × 12 = 96
9 × 12 = 108
10 × 12 = 120
11 × 12 = 132
12 × 12 = 144

There is an infinite number of multiples of 12. This is because we can continue adding 12 to find the next one.

How to Find the Multiples of 12

To find multiples of 12, multiply any whole number by 12. For example, the tenth multiple of 12 is 120 because 10 × 12 = 120. Alternatively, start from 0 and count up in twelves to find the multiples of 12.

Starting from 0 we can count up in twelves.

0 + 12 = 12. 12 + 12 = 24. 24 + 12 = 36. 36 + 12 = 48 and so on.

Here is a chart showing the multiples of 12 found by counting up in twelves each time.

To get from one multiple of 12 to the next, simply add 12.

chart showing the first multiples of 12 to 100

When teaching multiples of 12, it helps to look for patterns. We can remember that multiples of twelve repeat the pattern of ending in 2, 4, 6, 8 and 0. All multiples of 12 are even.

all multiples of 12 are even

When finding larger multiples of 12, it is quicker to simply multiply 12 by the multiple required.

To find the hundredth multiple of 12, multiply 12 by 100. 12 × 100 = 1200 and so, the hundredth multiple of 12 is 1200.

Multiples of 12 up to 100

There are 8 multiples of 12 that are less than 100. They are:

12, 24, 36, 48, 60, 72, 84 and 96.

Here is a chart showing all of the multiples of 12 up to 100.

multiples of 12 up to 100 chart

Here are the first 100 multiples of 12:

12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 528, 540, 552, 564, 576, 588, 600, 612, 624, 636, 648, 660, 672, 684, 696, 708, 720, 732, 744, 756, 768, 780, 792, 804, 816, 828, 840, 852, 864, 876, 888, 900, 912, 924, 936, 948, 960, 972, 984, 996, 1008, 1020, 1032, 1044, 1056, 1068, 1080, 1092, 1104, 1116, 1128, 1140, 1152, 1164, 1176, 1188 and 1200.

Now try our lesson on Finding Factors of a Number where we learn how to find factors.

2-Digit Column Addition

factors

Multiples of 12: Interactive Questions

Multiples of 12: Interactive Questions Return to video lesson on Multiples of 12

Multiples of 11

multiples of 11 poster

The multiples of 11 are numbers that can be divided exactly by 11, leaving no remainder.

The first few multiples of 11 are the numbers in the 11 times table.

For example 2 × 11 = 22 and so, the second multiple of eleven is 22.

For example 100 × 11 = 1100 and so, the hundredth multiple of 11 is 1100.

A multiple of 11 is any number that can be divided by 11 exactly, leaving no remainder.

Multiples of 11 Flashcards

Click on the multiples of 11 flashcards below to memorise the 11 times table:

Supporting Lessons

Multiples of 11: Interactive Questions

Multiples of 11 Worksheets and Answers