problem solving using factors and multiples

Factors and Multiples

Factors and multiples are different things.

But they both involve multiplication :

• Factors are what we can multiply to get the number
• Multiples are what we get after multiplying the number by an integer (not a fraction).

Example: the positive factors, and some multiples, of 6 :

• 1 × 6 = 6, so 1 and 6 are factors of 6
• 2 × 3 = 6, so 2 and 3 are factors of 6
• 0 × 6 = 0, so 0 is a multiple of 6
• 1 × 6 = 6, so 6 is a multiple of 6
• 2 × 6 = 12, so 12 is a multiple of 6

(Note: there are negative factors and multiples as well)

Here are the details:

"Factors" are the numbers we can multiply together to get another number:

A number can have many factors.

Example: 12

• 3 × 4 = 12, so 3 and 4 are factors of 12
• Also 2 × 6 = 12, so 2 and 6 are also factors of 12,
• And 1 × 12 = 12, so 1 and 12 are factors of 12 as well.

AND because multiplying negatives makes a positive , −1, −2, −3, −4, −6 and −12 are also factors of 12:

• (−1) × (−12) = 12
• (−2) × (−6) = 12
• (−3) × (−4) = 12

So ALL the factors of 12 are:

1, 2, 3, 4, 6 and 12 AND −1, −2, −3, −4, −6 and −12

Learn about Greatest Common Factor and how to find All Factors of a Number .

A multiple is the result of multiplying a number by an integer (not a fraction).

Example: Multiples of 3 are

..., −9, −6, −3, 0, 3, 6, 9, ...

Example: 15 is a multiple of 3, as 3 × 5 = 15 Example: 16 is not a multiple of 3

Example: Multiples of 5 are

..., −15, −10, −5, 0, 5, 10, 15, ...

Example: 10 is a multiple of 5, as 5 × 2 = 10 Example: 11 is not a multiple of 5

Multiples of Anything

We must multiply by an integer , but the number that is being multiplied can be anything.

Example: Multiples of π

..., −2 π , − π , 0, π , 2 π , 3 π , 4 π , ...

Factors And Multiples

Related Topics: More Lessons for Arithmetic Math Worksheets

If a is divisible by b , then b is a factor of a , and a is a multiple of b . For example, 30 = 3 × 10, so 3 and 10 are factors of 30 and 30 is a multiple of 3 and 10

Take note that 1 is a factor of every number.

Understanding factors and multiples is essential for solving many math problems.

Prime Factors

A factor which is a prime number is called a prime factor . For example, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5

You can use repeated division by prime numbers to obtain the prime factors of a given number.

Greatest Common Factor (GCF)

As the name implies, we need to list the factors and find the greatest one that is common to all the numbers. For example, to get the GCF of 24, 60 and 66:

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60

The factors of 66 are 1, 2, 3, 6, 11, 22,33 and 66

Look for the greatest factor that is common to all three numbers - thus 6 is the GCF of 24, 60 and 66.

Least Common Multiple (LCM)

As the name implies, we need to list the multiples and to find the least one that is common to all the numbers. For example, to get the LCM of 3, 6 and 9:

The multiples of 3 are 3, 6, 9, 12, 15, 18, 21 … The multiples of 6 are 6, 12, 18, 24, …

The multiples of 9 are 9, 18, 27, …

Look for the least multiple that is common to all three numbers - thus 18 is the LCM of 3, 6 and 9.

Shortcut To Finding LCM Here is a useful shortcut (also called the ladder method) to finding the LCM of a set of numbers. For example, to find the LCM of 3, 6 and 9, we divide them by any factor of the numbers in the following manner:

How to use the Ladder method to find GCF, LCM and simplifying fractions? Step 1: Write the two numbers on one line. Step 2: Draw the L shape. Step 3: Divide out common prime numbers starting with the smallest. LCM makes an L. GCF is down the left side. Simplified fraction is on the bottom. Example: Find the GCF, LCM and simplified fraction for 24 and 36.

LCM & GCF With the Ladder Method Example: Find the LCM and GCF of 24 and 36.

Difference between greatest common factor and least common multiple Example: Find the GCF and LCM of 16 and 24.

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Factors and Multiples

Factors and multiples are related to each other. A factor of a number is the number that divides it completely without leaving any remainder. For any given number we can represent it as p × q = z. Here we say z is a multiple of p and q. According to the definition of factors and multiples, p and q are factors of z, because z is divisible by p as well as q. For example, 6 × 2 = 12, so 6 and 2 are the factors of 12, and 12 is a multiple of 6 and 2.

How to Find Factors and Multiples?

To find factors and multiples of any given number, let say 'p', we have to find the list of numbers that divide the number 'p' without leaving any remainder. Here let us take an example of number 28. How can we find factors of 28 ?

Steps to Find Factors of a Number

• Step 1: List the two numbers such that their product gives 28.
• Step 2: Let's say we take 4 and 7 as the numbers.
• Step 3: Here 28 is divisible by 4 and 7 without leaving any remainder.
• Step 4: Hence, 28 can be factored in as 28 = 7 × 4.
• Step 5: Further, there are finite numbers by which a number can be divided without leaving any remainder . List out the numbers. Here, in the case of 28, the numbers are 1, 2, 4, 7, 14, and 28.
• Step 6: Factors of 28 can be listed as 1, 2, 4, 7, 14, and 28.

1 and the number itself are always the factors of the respective number.

Finding Multiples of a Number

Multiples of a number are the numbers that we get after multiplying the number by a whole number. Here, let us take the same example of number 28. How can we find multiples of 28 ? The multiples of 28 are all the numbers that result from the multiplication of 28 by another whole number. Let us look at the skip counting method shown in the image below. The skip counting method is one of the simplest methods to find the multiples of any given number.

Here, 28, 56, 84, 112, 140, and 168 are a few multiples of 28. There are infinite multiples of any given number.

Common Factors and Multiples

In this section, you will learn how to find common factors and multiples of any number. We know that a factor is a number that exactly divides the given number. Hence, a factor is nothing but a divisor of the given number. To find the factors, we can use the multiplication as well as the division method. To check if two or more numbers have common factors between them we can follow the below steps:

• Write the given numbers.
• Write all the possible factors of the given numbers
• Find the factors which are the same in both the numbers.
• Encircle the common factors.

This way we can easily find common factors between two or more numbers. Let us look at some examples.

Common factors of 30 and 42:

Therefore, the common factors of 30 and 42 are 1, 2, 3, and 6.

Here is another example with three numbers. The common factors of 42, 70, and 84 are shown in the image below.

Hence the common factors of 42, 70, and 84 are 1, 2, 7, and 14. Now let us look at ways to find common multiples.

There are various methods to find common multiples . Two of such ways are:

• Listing multiples method. In this, we list the multiples of each number and then we find their common multiples. It is one of the simplest methods by which we can find the common multiples of two or more numbers.
• Venn diagram method

Let us use the Venn diagram method. First, mark the multiples of any 2 numbers, let's say 3 and 4 in two separate circles. Look for the common numbers coming in both circles. The circle of multiples of 3 intersects with the circle of multiples of 4. The intersection part has the common multiples that belong to 3 as well as 4. Please note that there are infinite numbers of common multiples for every pair of numbers. The first three common multiples of 3 and 4 are 12, 24, and 36.

Now, let us look at an example of the listing multiples method. To find the common multiples of 2 and 4, we list their multiples and then find their common multiples.

• Multiples of 2 : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. 22, 24, etc.
• Multiples of 4 : 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, etc.
• Common multiples of 2 and 4: 4, 8, 12, 16, 20, 24, etc …

If a number is a factor of another number, then their common multiples are all the multiples of the larger number (as in the above example).

Properties of Factors and Multiples

There are many properties of factors and multiples that will help you to understand this concept properly. Those properties are listed below:

• 1 is the factor of every number.
• 0 is the multiple of every number.
• Factors and multiples are only used and applicable to whole numbers .
• For every number, 1 is the smallest factor and that number itself is the largest factor.
• Every number is a multiple of itself.
• There are an infinite number of multiples and a finite number of factors possible for every number.
• If there are only two factors of a number, i.e 1 and the number itself, that number is known as a prime number in math.

Important Notes

• Multiples of any number are infinite. Therefore, a set of any two numbers can have an infinite number of common multiples.
• The Least Common Multiple (LCM) is the smallest common multiple of sets of any two or more numbers. For example, the LCM of 3 and 4 is 12.

Related Articles on Factors and Multiples

Check these interesting articles related to the concept of factors and multiples grade 5.

• Prime Factorization

Examples of factors and Multiples

Example 1: Find all the numbers which are less than 100 and are common multiples of 3 and 4?

Solution: As per the definition of common factors and multiples, the common multiples of 3 and 4 are equal to the multiples of LCM of 3 and 4. The LCM of 3 and 4 is 12. All the multiples of 12 which are less than 100 are 12, 24, 36, 48, 60, 72, 84, and 96. Therefore, the common multiples of 3 and 4 are 12, 24, 36, 48, 60, 72, 84, and 96.

Example 2: What are the factors and multiples of 60?

The factors of 60 are all the whole numbers that can divide 60 without leaving any remainder. So, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. The mutiples of 60 can be found by multiplying 60 with whole numbers starting from 1. 60 × 1 = 60 60 × 2 = 120 60 × 3 = 180 60 × 4 = 240 60 × 5 = 300 So, the list of multiples of 60 includes 60, 120, 180, 240, 300, and so on. Therefore, the factors and multiples of 60 are: Factors = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 Multiples = 60, 120, 180, 240, 300, ...

Example 3: Find the common factors and multiples of 4, 8, and 12.

Solution: To find the common factors and multiples of 4, 8, and 12 we will use the listing method. Let us write separately factors of 4,8 and 12. There are 3 factors of 4 which are 1, 2, and 4. There are 4 factors of 8 listed as 1, 2, 4, and 8. There are 6 factors of 12 which are 1, 2, 3, 4, 6, and 12. The common factors of 4, 8, and 12 are 1, 2, and 4. Let us make a list of multiples of 4, 8, and 12. Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72 ... Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 ... Multiples of 12 = 12, 24, 36, 48, 60, 72, ... So, the common multiples of 4, 8, and 12 are 24, 48, 72, ...

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Practice Questions on Factors and Multiples

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FAQs on Factors And Multiples

What are factors and multiples.

Factors and multiples are two related concepts in math. A factor is a number that divides the given number exactly with 0 as the remainder. And a multiple is a number that is obtained by multiplying the given number with any whole number. For example, if it is given that 5 × 6 = 30. Here, 30 is the multiple of 5 and 6, and 5, and 6 are the factors of 30 .

What are the Factors and Multiples of 4?

The factors of 4 are 1, 2, and 4. And the multiples of 4 include 4, 8, 12, 16, 20, etc.

What is the Relationship Between Factors and Multiples?

The relationship between factors and multiples is that if we express it in the form of a multiplication fact "p × q = r", then the p and q represent the factors of the product. The product 'r' is the multiple of p and q.

What is the Difference Between Factors and Multiples of a Number?

The differences between factors and multiples are given below:

• Factors of a number are always less than or equal to the given number. While multiples of a number are always equal to or greater than the given number (except 0).
• A factor is a quantity in which we are dividing the given quantity, while multiple is the quantity that we are dividing into smaller units.

How to Teach Factors and Multiples to 4th Grade?

There are many activities and games for teaching factors and multiples to fourth graders. The activities are listed below:

• Play clap activity with learners. Choose any number, let's say 5, and ask learners to start counting from 1 such that one person will speak out one number only at their turn. Then, whoever will have the number 5 or multiples of 5 will not say the number. Instead, they have to clap one time. This way, we will get multiples of 5 .
• Give factors and multiples worksheets to learners.

How do you Use Factor and Multiples?

Factors and multiples are very commonly and actively used in mathematics. So, it is important to understand this concept in detail. It is used in division, measurements, patterns, etc.

What is a Factors and Multiples Formula?

An equation that represents both factors and multiples of a number is a × b = c, where c represents the multiple of a and b, and a, and b are the factors of c.

What are Common Factors and Multiples?

Common factors and multiples can be found using the listing method or by using prime factorization method. A common factor of two or more numbers is the number that can divide all those numbers excatly without leaving any remainder. For example, 1 and 5 are the common factors are 5, 15, and 25. A common multiple is a number that comes in the multiplication table of more than two numbers and hence, it is said to be the common multiple of the respective numbers. For example, the first two common multiples of 3 and 5 are 15 and 30.

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Understanding multiplication

Here you will learn about multiples, including how to calculate multiples of a number and solve problems using knowledge of multiples.

Students will first learn about multiples as part of operations and algebraic thinking in elementary school.

What are multiples?

Multiples are the products you get when two numbers, or factors, are multiplied.

For example, the first five multiples of 7 are 7 , 14 , 21 , 28 , and 35 .

You can skip count on a number line to find multiples of 7 .

The number line shows the first 7 multiples of 7 , but you could keep skipping counting to find more multiples.

To make them easier to find, multiples of the numbers 1 to 9 have patterns. They can help you decide if a given number is a multiple of 1 to 9 .

Common Core State Standards

How does this relate to 4th grade math?

• 4th Grade: Operations and Algebraic Thinking (4.OA.B.4) Determine whether a given whole number in the range 1 to 100 is a multiple of a given one-digit number.

How to identify multiples

In order to find if a number is a multiple of another number:

• Think about the pattern for the multiples of the number.

Decide if the number is a multiple.

[FREE] Multiples Worksheet (Grades 4 & 5)

Use this worksheet to check your grade 4 and 5 students’ understanding of multiples. 15 questions with answers to identify areas of strength and support!

Multiples examples

Example 1: decide whether a number is a multiple.

Is 76 a multiple of 6 ?

• Think about the pattern for the multiples of the number. 

Start a list of multiples for 6 :

6 , 12 , 18 , 24 , 30 …

Notice that the multiples of 6 are also multiples of 2 and 3 .

This pattern is true for all multiples of 6 .

2 Decide if the number is a multiple .

✔ The last digit (6) is even, so it is a multiple of 2 .

✘ The sum of the digits (7 + 6 = 13) , so it is not a multiple of 3 .

76 is NOT a multiple of 6 because it is NOT a multiple of 3 .

Skip counting on a number line shows that 76 is NOT a multiple of 6 .

Example 2: decide whether a number is a multiple

Is 65 a multiple of 5 ?

Think about the pattern for the multiples of the number .

Start a list of multiples for 5 :

5 , 10 , 15 , 20 , 25 …

Notice that they end in either a 5 or a 0 .

This pattern is true for all multiples of 5 .

✔ 65 is a multiple of 5 because it ends in a 5 .

Skip counting on a number line shows that 65 is the 13^{th} multiple of 5 .

Example 3: decide whether a number is a multiple

Is 66 a multiple of 4 ?

Start a list of multiples for 4 :

4 , 8 , 12 , 16 , 20 …

Notice that they are all doubles of a multiple of 2 .

This pattern is true for all multiples of 4 .

✘ 66 is the double of 33 \rightarrow 33 \times 2=66 , but 33 is not a multiple of 2 .

66 is not a multiple of 4 because it is not a double of a multiple of 2 .

Skip counting on a number line shows that 66 is NOT a multiple of 4 .

Example 4: decide whether a number is a multiple

Is 72 a multiple of 9 ?

Start a list of multiples for 9 :

9 , 18 , 27 , 36 , 45 …

Notice that the sum of the digits is divisible by 9 .

This pattern is true for all multiples of 9 .

✔ 72 is a multiple of 9 , because the sum of the digits (7+2=9) is divisible by 9 .

Skip counting on a number line shows that 72 is the eighth multiple of 9 .

Example 5: decide whether a number is a multiple

Is 42 a multiple of 3 ?

Start a list of multiples for 3 :

3 , 6 , 9 , 12 , 15 …

Notice that the sum of the digits is divisible by 3 .

This pattern is true for all multiples of 3 .

✔ 42 is a multiple of 3 , because the sum of the digits (4+2=6) is divisible by 3 .

Skip counting on a number line shows that 42 is the 14^{th} multiple of 3 .

Example 6: decide whether a number is a multiple

An elementary school is performing a play. They want to put out 88 chairs for the audience. If they want the same amount of chairs in each row, should they do rows of 6 , 8 or 10 ?

Multiples of 6 : the sum of the digits is divisible by three.

Multiples of 8 : the number is double a multiple of 4 .

Multiples of 10 : the last digit is zero.

✘ 88 is NOT a multiple of 6 , because the sum of the digits (8 + 8 = 16) is NOT divisible by 3 .

✔ 88 is a multiple of 8 , because it is double a multiple of 4 \rightarrow 2 \times 44=88 .

The school should put the 88 chairs in rows of 8 .

Teaching tips for multiples

• Use a multiplication table so that students can see examples of multiples and look for patterns.
• While students will need to remember the multiple patterns to solve quickly, avoid just using math worksheets. Help students deepen their understanding and likelihood of remembering by also doing hands-on activities. For example, give students 52 counters and ask them to make as many arrays as possible to find out what factors 52 is a multiple of.
• Play games regularly to help students memorize larger multiples. For example, have students count starting from 1 (the first student says 1 , the second says 2 , the third says 3 and so on), but challenge them to use a different word for a certain multiple each time. So if they were playing with multiples of 7 , whenever a student got to a multiple of 7 , they would say “cookie” instead of the multiple.

Easy mistakes to make

• Confusing the terms multiples and factors Factors and multiples are easily mixed up. Remember multiples are the numbers in the middle of the multiplication table, whereas factors are the numbers around the outside (the numbers being multiplied).

• Forgetting the number itself as a multiple All numbers are a multiple of themselves. For example, the multiples of 6 are: 6 , 12 , 18 , 24 and so on and so 6 is a multiple of itself.

Related lessons on factors and multiples

• Factors and multiples
• Least common multiple
• Greatest common factor
• Factor tree
• GCF and LCM
• Prime factors
• Factor pairs

Practice multiples questions

1. Which number is a multiple of 2 ?

For multiples of 2 , the last digit is an even number. 54 ends in an even number and 27 \times 2=54 , so 54 is a multiple of 2 .

2. Which number is a multiple of 7 ?

Using a number line to skip count by 7 , shows that 91 is a multiple of 7 .

3. Which number is NOT a multiple of 3 ?

For multiples of 3 , the sum of the digits is divisble by three. 4+3=7 is not divisible by 3 , so 43 is NOT a multiple of 3 .

4. Which number is NOT a multiple of 4 ?

For multiples of 4 , the number is double a multiple of 2 . 58 is the double of 29 , so 58 is NOT a multiple of 4 .

5. A grocery store has 78 boxes of cereal. They want to put the same number of boxes on each shelf. How many boxes should they put on each shelf?

For multiples of 6 , the number is divisible by both two and three. 78 is a multiple of 2 , because it ends in an even number (8) . 78 is a multiple of 3 , because 7 + 8 = 15 and 15 \div 3 = 5 .

So 78 is a multiple of 6 \rightarrow 6 \times 13=78 The grocery store can put 6 boxes on each shelf.

Multiples FAQs

Times tables are a list of multiplication equations for a certain number and include both factors and multiples. Multiples are the answers (products) in times tables.

Yes, if a number divides another number without a remainder, it’s a multiple. For example, in the question “is 72 a multiple of 2 ?” you can solve 72 \div 2=36 . There are no remainders, so it is a multiple.

All positive and negative numbers have multiples. However, in 4th grade math you only work with positive numbers.

Prime numbers have an infinite amount of multiples but only 2 factors.

The next lessons are

• Converting fractions, decimals and percentages
• Fractions operations

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Factors and Multiples Worksheets

Welcome to our Factors and Multiples Worksheets. Here you will find a wide range of free Math Worksheets which will help your child to learn to use multiples and factors at a 4th Grade/ 5th Grade level.

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Factors and Multiples Worksheet

These sheets have been designed to support your child with their learning of multiples and factors.

The sheets are graded in order of difficulty with the easiest sheet coming first in each section.

Using these sheets will help your child to:

• Know and understand what multiples and factors are;
• apply knowledge of multiples and factors to solve problems;
• Develop and practice their mental calculation skills.

Want to test yourself to see how well you have understood this skill?.

• Try our NEW quick quiz at the bottom of this page.

Quicklinks to ...

Multiples and Factors Help

Multiples worksheets, factors worksheets.

• Factors and Multiples Riddles
• Easier/Harder Worksheets
• More related resources

Factors and Multiples Online Quiz

If a number is a multiple of another number, it means that it can be made out of adding groups of the other number together.

12 is a multiple of 4 because 4 + 4 + 4 = 12 (or 4 x 3 = 12)

27 is a multiple of 9 because 9 + 9 + 9 = 27 (or 9 x 3 = 27)

17 is not a multiple of 4 because it cannot be made by adding groups of 4 together.

A factor is a number that divides into another number with no remainder.

In other words every number is divisible by each of its factors.

1 is a factor of every whole number, because every integer is divisible by one.

3 and 7 are both factors of 21 because 3 x 7 = 21

10 and 6 are both factors of 60 because 10 x 6 = 60

7 is not a factor of 24 because 24 is not divisible by 7 (24 ÷ 7 = 3 remainder 3).

Multiples and Factors are connected with each other:

• if we know that 3 is a factor of 12, then 12 is a multiple of 3
• if we know that 33 is a multiple of 11, then 11 is a factor of 33.
• also, if we know that 24 is not a multiple of 7, then 7 is not factor of 24.

The example below shows the relationship visually.

If we know that 3 is a factor of 24, then 24 must be a multiple of 3.

If we know that 24 is a multiple of 3, then 3 must be a factor of 24.

Multiples and Factors Worksheets

We have split our worksheets into 3 different sections:

• the first section contains only worksheets about Multiples
• the second section contains only worksheets about Factors
• the third section contains worksheets with both Factors and Multiples
• Multiples Sheet 4:1
• PDF version
• Multiples Sheet 4:2
• Multiples Sheet 4:3
• Multiples Sheet 4:4

We have two worksheets on finding Factor Pairs up to 100.

We have two worksheets which involve finding all the factors of different numbers.

• Factor Pairs Worksheet 1
• Factor Pairs Worksheet 2
• Factors Worksheet 4:1
• Factors Worksheet 4:2
• Factors and Multiples Worksheet 4:1
• Factors and Multiples Worksheet 4:2
• Factors and Multiples Worksheet 4:3

Factor and Multiples Riddles

Using riddles is a great way to get children to apply their knowledge of factors and multiples to solve problems.

It is also a good way to get children working collaboratively and talking about the language together.

Each riddle consists of some clues and a selection of possible answers.

Solving the clues gradually eliminates all the incorrect answers leaving just one solution.

• Factors and Multiples Riddles 1
• Factors and Multiples Riddles 2

Looking for some easier Multiples Sheets

The sheets in this section cover similar areas to the worksheets on this page but are at an easier level.

• round a number to the nearest 10, 100 or 1000;
• use the > and < symbols correctly for inequalities;
• use multiples and apply them to solve problems.
• Rounding Inequalities Multiples Worksheets

Looking for some harder Factors and Multiples Worksheets

We also have some more advanced worksheets about multiples and factors.

The worksheets below are more suitable for 6th graders and above.

• Greatest Common Factor Worksheets
• Least Common Multiple Worksheets
• Factor Tree Worksheets (easier)
• Prime Factorization Worksheets (harder)

More Recommended Math Resources

Take a look at some more of our worksheets similar to these.

Divisibility Rules 1-10 Chart

We have a range of charts which can help you determine whether a number between 1 and 10 is a factor of a number.

• Divisibility Rules 1-10 Charts

Balancing Math Equations Worksheets

The sheets in this area will help your child understand the use and purpose of the equals sign (=) in an equation.

It will also help children learn to start manipulating and calculating numerical expressions so that they are equivalent.

This will stand them in good stead for when they start to learn algebra, and manipulate algebraic equations.

• Balancing Math Equations

Sieve of Erastosthenes

The Sieve of Erastosthenes is a method for finding what is a prime numbers between 2 and any given number.

Eratosthenes was a Greek mathematician (as well as being a poet, an astronomer and musician) who lived from about 276BC to 194BC.

If you want to find out more about his sieve for finding primes, and print out some Sieve of Eratosthenes worksheets, use the link below.

• Sieve of Eratosthenes page

Want to find out more about primes?

Take a look at our Prime Number page which clearly describes what a prime numbers is and what they are not.

There are also many different questions about prime numbers answered, as well as information about the density of primes.

• What is a Prime Number

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39.3: Using Common Multiples and Common Factors

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Let's use common factors and common multiple to solve problems.

Exercise \(\PageIndex{1}\): Keeping a Steady Beat

Your teacher will give you instructions for playing a rhythm game. As you play the game, think about these questions:

• When will the two sounds happen at the same time?
• How does this game relate to common factors or common multiples?

Exercise \(\PageIndex{2}\): Factors and Multiples

Work with your partner to solve the following problems.

• Find a number of plates and cups that meets her requirement.
• How many packs of each supply will she need to buy to get that number?
• Name two other quantities of plates and cups she could get to meet her requirement.
• What is the largest possible tile size she could use? Write the side length in feet. Explain how you know it’s the largest possible tile.
• How many of these largest size tiles are needed?
• Name more tile sizes that are whole number of feet that she could use to cover the bathroom floor. Write the side lengths (in feet) of the square tiles.
• Name three lockers that will get both stickers.
• After Lin makes her way down the hall, will the 30th locker have no stickers, 1 sticker, or 2 stickers? Explain how you know.
• What is the largest number of kits the nurse can make?
• How many bandages and lozenges will be in each kit?
• What kind of mathematical work was involved in each of the previous problems? Put a checkmark to show what the questions were about.

Are you ready for more?

You probably know how to draw a five-pointed star without lifting your pencil. One way to do this is to start with five dots arranged in a circle, then connect every second dot.

If you try the same thing with six dots arranged in a circle, you will have to lift your pencil. Once you make the first triangle, you’ll have to find an empty dot and start the process over. Your six-pointed star has two pieces that are each drawn without lifting the pencil.

With twelve dots arranged in a circle, we can make some twelve-pointed stars.

• Start with one dot and connect every second dot, as if you were drawing a five-pointed star. Can you draw the twelve-pointed star without lifting your pencil? If not, how many pieces does the twelve-pointed star have?

• This time, connect every third dot. Can you draw this twelve-pointed star without lifting your pencil? If not, how many pieces do you get?

• What do you think will happen if you connect every fourth dot? Try it. How many pieces do you get?

• Do you think there is any way to draw a twelve-pointed star without lifting your pencil? Try it out.

• Now investigate eight-pointed stars, nine-pointed stars, and ten-pointed stars. What patterns do you notice?

Exercise \(\PageIndex{3}\): More Factors and Multiples

Here are five more problems. Read and discuss each one with your group. Without solving , predict whether each problem involves finding common multiples or finding common factors. Circle one or more options to show your prediction.

• Soccer. Diego and Andre are both in a summer soccer league. During the month of August, Diego has a game every 3rd day, starting August 3rd, and Andre has a game every 4th day, starting August 4th.
• common multiples
• least common multiple
• common factors
• greatest common factor
• What is the first date that both boys will have a game?
• How many of their games fall on the same date?
• Performances. During a performing arts festival, students from elementary and middle schools will be grouped together for various performances. There are 32 elementary students and 40 middle-school students. The arts director wants identical groups for the performances, with students from both schools in each group. Each student can be a part of only one group.
• Name all possible groupings.
• What is the largest number of groups that can be formed? How many elementary school students and how many middle school students will be in each group?
• Lights. There is a string of holiday lights with red, gold, and blue lights. The red lights are set to blink every 12 seconds, the gold lights are set to blink every 8 seconds, and the blue lights are set to blink every 6 seconds. The lights are on an automatic timer that starts each day at 7:00 p.m. and stops at midnight.
• After how much time with all 3 lights blink at the exact same time?
• How many times total will this happen in one day?
• Banners. Noah has two pieces of cloth. He is making square banners for students to hold during the opening day game. One piece of cloth is 72 inches wide. The other is 90 inches wide. He wants to use all the cloth, and each square banner must be of equal width and as wide as possible.
• How wide should he cut the banners?
• How many banners can he cut?
• Dancers. At Elena’s dance recital her performance begins with a line of 48 dancers that perform in the dark with a black light that illuminates white clothing. All 48 dancers enter the stage in a straight line. Every 3rd dancer wears a white headband, every 5th dancer wears a white belt, and every 9th dancer wears a set of white gloves.
• If Elena is the 30th dancer, what accessories will she wear?
• Will any of the dancers wear all 3 accessories? If so, which one(s)?
• How many of each accessory will the dance teacher need to order?
• Your teacher will assign your group a problem. Work with your group to solve the problem. Show your reasoning. Pause here so your teacher can review your work.
• Work with your group to create a visual display that includes a diagram, an equation, and a math vocabulary word that would help to explain your mathematical thinking while solving the problem.
• Prepare a short presentation in which all group members are involved. Your presentation should include: the problem (read aloud), your group's prediction of what mathematical concept the problem involved, and an explanation of each step of the solving process.

Exercise \(\PageIndex{4}\): Factors and Multiples Bingo

Your teacher will explain the directions for a bingo game. Here are some things to keep in mind:

• Share one bingo board and some bingo chips with a partner.
• To play the game, your teacher will read statements aloud. You may help one another decide what numbers fit each statement, but speak only in a whisper. If the teacher hears anything above a whisper, you are out.
• The first person to call bingo needs to call out each number and identify the statement that it corresponds to. If there is an error in identifying statements, that player is out and the round continues.

Good luck, and have fun!

If a problem requires dividing two whole numbers by the same whole number, solving it involves looking for a common factor. If it requires finding the largest number that can divide into the two whole numbers, we are looking for the greatest common factor .

Suppose we have 12 bagels and 18 muffins and want to make bags so each bag has the same combination of bagels and muffins. The common factors of 12 and 18 tell us possible number of bags that can be made.

The common factors of 18 are 1, 2, 3, and 6. For these numbers of bags, here are the number of bagels and muffins per bag.

• 1 bag: 12 bagels and 18 muffins
• 2 bags: 6 bagels and 9 muffins
• 3 bags: 4 bagels and 6 muffins
• 6 bags: 2 bagels and 3 muffins

We can see that the largest number of bags that can be made, 6, is the greatest common factor.

If a problem requires finding a number that is a multiple of two given numbers, solving it involves looking for a common multiple. If it requires finding the first instance the two numbers share a multiple, we are looking for the least common multiple .

Suppose forks are sold in boxes of 9 and spoons are sold in boxes of 15, and we want to buy an equal number of each. The multiples of 9 tell us how many forks we could buy, and the multiples of 15 tell us how many spoons we could buy, as shown here.

• Forks: 9, 18, 27, 36, 45, 54, 63, 72, 90. . .
• Spoons: 15, 30, 45, 60, 75, 90. . .

If we want as many forks as spoons, our options are 45, 90, 135, and so on, but the smallest number of utensils we could buy is 45, the least common multiple. This means buying 5 boxes of forks (\(5\cdot 9=45\)) and 3 boxes of spoons (\(3\cdot 15=45\)).

Glossary Entries

Definition: Common Factor

A common factor of two numbers is a number that divides evenly into both numbers. For example, 5 is a common factor of 15 and 20, because \(15\div 5=3\) and \(20\div 5=4\). Both of the quotients, 3 and 4, are whole numbers.

• The factors of 15 are 1, 3, 5 , and 15.
• The factors of 20 are 1, 2, 4, 5 , 10, and 20.

Definition: Common Multiple

A common multiple of two numbers is a product you can get by multiplying each of the two numbers by some whole number. For example, 30 is a common multiple of 3 and 5, because \(3\cdot 10=30\) and \(5\cdot 6=30\). Both of the factors, 10 and 6, are whole numbers.

• The multiples of 3 are 3, 6, 9, 12, 15 , 18, 21, 24, 27, 30 , 33 . . .
• The multiples of 5 are 5, 10, 15 , 20, 25, 30 , 35, 40 . . .

The common multiples of 3 and 5 are 15, 30, 45, 60 . . .

Definition: Greatest Common Factor

The greatest common factor of two numbers is the largest number that divides evenly into both numbers. Sometimes we call this the GCF. For example, 15 is the greatest common factor of 45 and 60.

• The factors of 45 are 1, 3, 5, 9, 15 , and 45.
• The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15 , 20, 30, and 60.

Definition: Least Common Multiple

The least common multiple of two numbers is the smallest product you can get by multiplying each of the two numbers by some whole number. Sometimes we call this the LCM. For example, 30 is the least common multiple of 6 and 10.

• The multiples of 6 are 6, 12, 18, 24, 30 , 36, 42, 48, 54, 60 . . .
• The multiples of 10 are 10, 20, 30 , 40, 50, 60, 70, 80 . . .

Exercise \(\PageIndex{5}\)

Mai, Clare, and Noah are making signs to advertise the school dance. It takes Mai 6 minutes to complete a sign, it takes Clare 8 minutes to complete a sign, and it takes Noah 5 minutes to complete a sign. They keep working at the same rate for a half hour.

• Will Mai and Clare complete a sign at the same time? Explain your reasoning.
• Will Mai and Noah complete a sign at the same time? Explain your reasoning.
• Will Clare and Noah complete a sign at the same time? Explain your reasoning.
• Will all three students complete a sign at the same time? Explain your reasoning.

Exercise \(\PageIndex{6}\)

Diego has 48 chocolate chip cookies, 64 vanilla cookies, and 100 raisin cookies for a bake sale. He wants to make bags that have all three cookie flavors and the same number of each flavor per bag.

• How many bags can he make without having any cookies left over?
• Find the another solution to this problem.

(From Unit 7.4.1)

Exercise \(\PageIndex{7}\)

• Find the product of 12 and 8.
• Find the greatest common factor of 12 and 8.
• Find the least common multiple of 12 and 8.
• Find the product of the greatest common factor and the least common multiple of 12 and 8.
• What do you notice about the answers to question 1 and question 4?
• Choose 2 other numbers and repeat the previous steps. Do you get the same results?

Exercise \(\PageIndex{8}\)

• Given the point \((5.5,-7)\), name a second point so that the two points form a vertical segment. What is the length of the segment?
• Given the point \((3,3.5)\), name a second point so that the two points form a horizontal segment. What is the length of the segment?

(From Unit 7.3.1)

Exercise \(\PageIndex{9}\)

Find the value of each expression mentally.

• \(\frac{1}{2}\cdot 37-\frac{1}{2}\cdot 7\)
• \(3.5\cdot 40+3.5\cdot 60\)
• \(999\cdot 5\)

(From Unit 6.2.4)

Factors and Multiples

Factors are parts of numbers that, multiplied together, give a larger number. Every number has at least two factors, one and the number itself. On the other hand, multiples are numbers where the same number is repeated, as if you were counting by that number. For example, the multiples of 2 start with 2 and are: 2, 4, 6, 8, 10 . . . and so on. Each additional number is a multiple of 2.

Factors and multiples are especially important in working with expanding and reducing fractions, as well as finding patterns in numbers. Finding the greatest common factor , least common multiple , and prime factors of a number are important skills you will learn in this section.

• solve a system of inequalities graphically and determine one possible solution (give the 2 inequalities to graph)
• Can someone help me, please.
• Brittany can make 7 friendship bracelets in 2 hours. At this rate , how many hours will it take her to make 84 friendship bracelets
• Loga help please

Factors and Multiples – Definition, Differences, Examples, FAQs

What are factors and multiples, difference between factors and multiples, how to find factors and multiples, solved examples on factors and multiples, practice problems on factors and multiples, frequently asked questions about factors and multiples.

Factors and multiples are two interrelated concepts in mathematics. If $A \times B = C$ , then A and B are factors of C, whereas C is a multiple of both A and B.

Consider an example to understand.

Example: $3 \times 7 = 21$

Here, 3 and 7 are factors of 21.

21 is the multiple of both 3 and 7.

What Are Factors?

A factor is a number that divides the given number exactly, without any remainder. When a number n is divisible by b, we can say that b is a factor of n.

For example, the factors of 16 are 1, 2, 4, 8, and 16, because these numbers divide 16 exactly, without leaving any remainder.

Related Worksheets

Properties of Factors

• 1 is the smallest factor of any number.
• The number itself is the greatest factor of any number.
• A factor of a number is always less than or equal to the number.
• Factors of a number are finite.
• 0 cannot be a factor of any number.
• If B is a multiple of A, then A is a factor of B.

What Are Multiples?

A multiple of a number is the number obtained when the given number is multiplied with an integer. 

A multiple is a result of multiplying a number by an integer . Note that, when we study multiples of any number, we generally talk about positive multiples only (excluding 0 and negative multiples). Thus, we can say that a multiple is the number obtained by multiplying a given number by a positive integer. 

For example, the multiples of 5 are 5, 10, 15, 20,…, and so on, because these numbers can be obtained by multiplying 5 with the whole numbers 1, 2, 3, 4,…, and so on.

You can find the multiples of a number in the multiplication table (times table). Take a look!

$5 \times 1 = 5$

$5 \times 2 = 10$

$5 \times 3 = 15$

$5 \times 4 = 20$

$5 \times 5 = 25$

$5 \times 6 = 30$

$5 \times 7 = 35$

$5 \times 8 = 40$

$5 \times 9 = 45$

$5 \times 10 = 50$

Properties of Multiples

• Infinitely many multiples of any number.
• A multiple is always greater than or equal to the given number.
• Every number is a multiple of 1.
• 0 is considered a multiple of every number.

Let’s move ahead and learn the methods we can use to find factors and multiples of a number.

How can we find factors?

To find the factors of a given number, divide the number by each integer between 1 and the number itself. The integers that give us a remainder of 0 are factors of the number. 

For example, to find the factors of 6, divide 6 by numbers 1 to 6.

The factors of 6 are 1, 2, 3, and 6.

How to find multiples

To find the multiples of a given number, multiply the number by each positive integer.

For example, to find the multiples of 5, you would multiply 5 by 1, 2, 3, 4, and so on.

Multiples of $5 = 5,\; 10,\; 15,\; 20,\; 25$, …

Multiples of $9 = 9\;, 18\;, 27\;, 36\;, 45$, …

Least Common Multiple (LCM)

LCM of two numbers is the smallest positive integer that is a multiple of both the numbers. It is the smallest number among the common multiples of the given two numbers.

Let’s find the LCM of 4 and 6. 

Common multiples $= 12,\; 24,\; 36$, …

Least common multiple $= 12$

LCM (4, 6) $= 12$

Greatest Common Factor (GCF)

It is the largest positive integer that divides the given two integers without leaving any remainder. In other words, it is the largest number that is a factor of all the given integers.

Let’s find the GCD of 12 and 18.

The largest number that appears in both lists is 6. 

GCF of 12 and 18 is 6.

Facts about Factors and Multiples

• Every number has at least two factors, which are 1 and itself.
• A number can have an infinite number of multiples, but only a finite number of factors.
• Factors and multiples can be used to find the patterns and symmetries in nature, such as the arrangement of petals in flowers and the formation of crystals.
• Prime numbers have only two factors, 1 and itself, while composite numbers have more than two factors.
• The number 1 is neither prime nor composite. It only has one factor, which is itself.
• 0 is neither prime nor composite.

In this article, we have learned about factors and multiples, the definitions, how to find factors and multiples, and also the properties of multiples and factors of a number. We have also learned some interesting facts about factors and multiples. Let’s now look at some solved examples and practice problems to understand the concept better.

• Find the factors of 30.

Solution: 

The factors of 30 are the integers that divide 30 without leaving any remainder.

$30 \div 1 = 30 \text{and remainder} = 0$

$30 \div 2 = 15 \text{and remainder} = 0$

$30 \div 3 = 10 \text{and remainder} = 0$

$30 \div 5 = 6 \text{and remainder} = 0$

$30 \div 6 = 5 \text{and remainder} = 0$

30 \div 10 = 3 \text{and remainder} = 0$

$30 \div 15 = 2 \text{and remainder} = 0$

$30 \div 30 = 1 \text{and remainder} = 0$

Thus, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

• Find the multiples of 7 that are less than or equal to 56.

We can write the multiples of 7 as,

$7 \times 1 = 7$

$7 \times 2 = 14$

$7 \times 3 = 21$

$7 \times 4 = 28$

$7 \times 5 = 35$

$7 \times 6 = 42$

$7 \times 7 = 49$

$7 \times 8 = 56$

$7 \times 9 = 63$

Hence, the multiples of 7 that are less than or equal to 56 are 7, 14, 21, 28, 35, 42, 49, and 56.

• Find the least common multiple (LCM) of 8 and 12.

Multiples of 8: 8, 16, 24, 32, 40, 48…

Multiples of 12: 12, 24, 36, 48…

Common multiples: 24, 48…

LCM of 8 and 12 is the smallest number that is a multiple of both 8 and 12, which is 24.

• Find the greatest common factor (GCF) of 18 and 48.

Factors of 18 are 1, 2, 3, 6, 9, and 18. 

Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. 

Hence, the GCF of 18 and 48 is the greatest number that is a factor of both 18 and 24, which is 6.

• If 20 and 35 are both factors of a number n, what is the smallest possible value of n?

The smallest possible value of n is the least common multiple (LCM) of 20 and 35.

The multiples of 20 are 20, 40, 60, 80, 100, 120, 140 , 160, 180… and so on.

The multiples of 35 are 20, 40, 60, 80, 100, 120, 140 , 160, 180… and so on.

LCM of 20 and $35 = 140$.

Hence, the smallest possible value of n is 140.

Factors and Multiples - Definition, Differences, Examples, FAQs

Attend this quiz & Test your knowledge.

Which of the following is a factor of 27?

What is the greatest common factor of 12 and 36, which of the following numbers is a multiple of 5, the lcm of 4 and 7 is ______., what is the sum of the first five multiples of 3.

What are prime factors?

Prime factors are the factors of a number that are prime numbers. For example, the prime factors of 6 are 2 and 3 since 6 can be written as $2 \times 3$.

What is the difference between factors and multiples?

A factor of a number is a number that divides it evenly, while a multiple of a number is the result of multiplying that number by any whole number.

What is the method to find the factors of a large number?

To find the factors of a large number, you can start by dividing it by 2, then by 3, and so on, until you reach a prime number.

What is the difference between GCF and LCM?

The GCF is the largest factor that two or more numbers have in common, while the LCM is the least (smallest) multiple that two or more numbers have in common.

What are the uses of factors and multiples in real life?

Factors and multiples can be used in many real-life situations, such as finding the number of seats in a theater or the number of tiles needed to cover a floor. They are also important in mathematical concepts like fractions, ratios, and proportions.

How are factors and multiples related?

When we multiply two numbers to find the product, each number being multiplied is a factor of the product, whereas the number obtained from the product is a multiple of each number being multiplied.

What is the definition of factors and multiples?

Factor is a number that divides another number evenly (with remainder $=$ 0). Multiples are numbers obtained by multiplying a given number by a positive integer.

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Factors And Multiples

In the elementary level, the factors and multiples are the two key concepts which are studied together. Factors are the numbers which divide the given number exactly, whereas the multiples are the numbers which are multiplied by the other number to get specific numbers. It is time to recollect, to understand the concept of multiples and factors of a number.  In this article, you will learn the detailed definition of factors and multiples with examples and the difference between them in detail.

Table of Contents:

• Factor Definition
• Multiples Definition

Difference Between Factors and Multiples

What are the factors.

When a number is said to be a factor of any other second number, then the first number must divide the second number completely without leaving any remainder. In simple words, if a number (dividend) is exactly divisible by any number (divisor), then the divisor is a factor of that dividend. Every number has a common factor that is one and the number itself.

For example, 4 is a factor of 24, i.e. 4 divides 24 exactly giving 6 as quotient and leaving zero as remainder. Alternatively, 6 is also a factor of 24 as it gives 4 as quotient on division. Therefore, 24 has 1, 24, 4, 6 as its factors in addition to 2, 3, 8 and 12 and all these numbers divide 24 exactly leaving no remainder.

If any natural number has only two factors, i.e. 1 and the number itself as its factors, such numbers are called prime numbers . 2 is an example of a prime number where it has only two factors, i.e. 1 and 2.

What are Multiples?

A multiple of a number is a number that is the product of a given number and some other natural number. Multiples can be observed in a multiplication table. Multiples of some numbers are as follows:

Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on Hence, multiples of 2 will be even numbers and will end with 0, 2, 4, 6 or 8.

Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, and so on.

Multiples of 5 are 5, 10, 15, 20, 25, and so on.

Every multiple of 5 has its last digit as 0 or 5.

In the above-mentioned examples, say multiples of 2, the number 2 can be multiplied by infinite numbers to find the “n” number of multiples. Now, let us assume an example,

Here, 3 and 4 are the factors of 12,

12 is multiple of 3 and 4

Thus, we can conclude that if X and Y are two numbers and;

•    If X divides Y, X is a factor of Y

•    If Y is divisible by X, Y is a multiple of X

Since the number 1 divides every integer, it is a common factor of every integer. Also, every number is divisible by 1 and every number is a multiple of 1.

The difference between factors and multiples are given here in tabular form. Go through the below table to understand the difference between them.

Factors and Multiples Examples

Example 1:  

Find the factors of 20.

The factors of 20 are 1, 2, 4, 5, 10 and 20.

Because the number 20 is exactly divisible by these numbers leaving the remainder zero.

Example 2: 

Find the multiples of 20.

The multiples of 20 are 20, 40, 60, 80, 100, 120, …

20 × 1 = 20

20 × 2 = 40,

20 × 4 = 80

Frequently Asked Questions on Factors and Multiples

Mention the factors of 36..

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Explain the factors and numbers.

A factor is a number that divides the given number exactly without a remainder, whereas a multiple is a number, and that number is a product of the given number with other numbers.

Write down the common multiples of 3 and 7.

The common multiples of 3 and 7 are 21, 42, and 63.

Define the proper factor.

The proper factor is the factor of the number except 1 and the number itself. For example, the factors of 15 are 1, 3, 5, and 15. Then, the proper factors of 15 are 3 and 5

What is the smallest factor of 9?

The factors of 9 are 1, 3 and 9. Then the smallest factor of 9 is 1.

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What is multiples of 6548

There are infinite multiples of a number and are found by multiplying the number by natural numbers: 6548 X 1 = 6548 6548 X 2 = 13,096 6548 X 3 = 19,644 and so on

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Factors and Multiples Game

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Factors and Multiples printable worksheet - game Factors and Multiples printable worksheet - puzzle On this  resources page there are printable 1-100 square grids.

You may like to have a go at Dozens before having a go at this challenge.  This is a game for two players. The first player chooses a positive even number that is less than 50, and crosses it out on the grid. The second player chooses a number to cross out. The number must be a factor or multiple of the first number. Players continue to take it in turns to cross out numbers, at each stage choosing a number that is a factor or multiple of the number just crossed out by the other player.

The first person who is unable to cross out a number loses.

Play a few times to get a feel for the game.

Do you have any winning strategies?

Here is an interactive version of the game in which you drag the numbers from the left hand grid and drop them on the right hand grid. Alternatively, click on a number in the left hand grid and it will transport to the earliest empty location in the right hand grid. You can rearrange the numbers in the right hand grid by dragging and dropping them in position. The integer in the top right hand corner grows with the number of factors/multiples you have in a row.

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This activity featured in an NRICH video in June 2020 and an NRICH student webinar in November 2020.  

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Real-life Applications of Factors and Multiples

In the field of mathematics, factors and multiples play a crucial role in solving problems and understanding numerical relationships. In this article, we will explore the tangible applications of factors and multiples in everyday situations. Through examples, we will illustrate how factors and multiples serve as useful tools in problem-solving across various domains.

What are the Factors?

Factors are numbers that divide a given quantity without leaving a remainder.

In real-life scenarios, factors often represent essential components or divisible units. For instance, when considering the factors of a number like 12, which are 1, 2, 3, 4, 6, and 12, each factor represents a potential division or subdivision of the original quantity.

What are Multiples?

Multiples are numbers that are the product of multiplying a given quantity by an integer.

In practical terms, multiples represent repetitions or extensions of a base quantity. For instance, multiples of 5 include 5, 10, 15, 20, and so on, indicating the successive multiplication of 5 by integers.

Factors and multiples are cornerstones of mathematics and are employed as a tool in numerous daily situations. Real-life applications of factors and multiples are:

Financial Valuation and Investment Strategy

As the basis of financial valuation techniques – among DCF and CCA – factors and multiples are the main drivers. Analysts use the companies’ financial performance as the basis for judgement by applying multiples such as earnings multiples and revenue multiples, guiding investment decisions and portfolio management strategies.

• Example: For equity valuation analysis, an analyst first determines the price-to-earnings (P/E) ratio of a firm by dividing the stock price by the earnings per share (EPS). A sink P/E ratio means the stock is undervalued while a high P/E ratio signals stock overvaluation, thus prompt investors to pay attention to stock selection and asset allocation.

Production Efficiency and Resource Utilization

Factors and multiples are used to create a production plan for the best use of the resources and for minimizing the production costs. Industrialists calculate production division measurements such as machine cycle time multiples and setup time multiples in order to pinpoint production bottlenecks , streamline production processes and improve overall productivity.

• Example: The production managers in a factory need to analyze machine cycle time multiples in order to identify which machines have longer cycle times, creating a schedule for the production accordingly. In a situation where there is not much setup time multiples and high machine utilization, manufacturers are able to increase productivity and decrease manufacturing lead times.

Infrastructure Resilience and Safety Standards

Factors and multiples are of great importance in the sense these things assure the sustainability and safety of infrastructure endeavors. Engineers review the structural safety factors and load multiples for designing buildings, bridges, and other similar structures that can take all kind of external forces, environmental conditions, or potential hazards.

Example: In bridge engineering, structural engineers compute and factor the safety for components such as beams and columns to ensure that they can bear design load, live load and dynamic load such as wind and seismic forces. Through the adherence to the safety regulations and the legislation of design mathematical formulas, engineers minimize possible structural failures and maintain public safety.

Supply Chain Flexibility and Risk Mitigation

Factors and multiples are addressed in supply chain management to measure supply chain adaptability, flexibility, and resiliency. Supply chain managers measure multiples of lead time and demand variability to uncover supply chain vulnerabilities ,in addition, they create contingency plans and help to mitigate supply chain risks.

• Example: In supply chain management, supply chain managers apply lead time multiples to find out minimum levels of safety stock and product reorder points of the critical inventory items. While ensuring safety stock and adjusting inventory levels in time to account for lead time variation, enterprises can avoid stockouts, disrupt routine processes in the supply chain, and preserve customer service levels.

Data-driven Decision Making and Predictive Analytics

Factor and multiples play a critical role in the processes of data-driven decision-making involving categorization of patterns, correlations, and cause and effect relationships in sets of data. Data analysts use tools such as common factors and least common multiples to gain meaningful insights to develop predictive models and business process optimization.

• Example: To find the common patterns in a retail market, analysts employ the technique of market basket analysis which includes examining what customers often purchase and product correlations. With the knowledge of LCM, retailers can predict cross-sell probabilities, assemble appropriate range, and create better customer experiences.

Educational Pedagogy and Learning Strategies

Among the essential ideas in mathematics, factors and multiples play a vital role, for they are crucial to the development of learners’ critical thinking skills, ability to solve problems and mathematical literacy. Education is the medium through which factors and multiples are employed to reveal the underlying aspects of prime factorizations, least common multiples, and greatest common divisors that again improve conceptual knowledge coupled with mathematical proficiency.

• Example: Through real-time instruction, teachers conduct exercises and use manipulatives to help students embrace the idea of understanding of factors and multiples conceptually and logically. The close-to-real resemblance and problem-solving scenarios examples which teachers use are useful in the students’ development of mathematical reasoning skills and application of mathematical concepts to everyday matters.

Transportation and Logistics

In transport and logistics operations, Factors and Multiples is in charge of route optimization, vehicles scheduling and cargo consolidation. Transportation planners make use of parameters like distance multiples, vehicle capacity multiples to create the most cost effective transport, fuel consumption and improve delivery efficiency, among other things.

• Example: In freight transport, logistics companies use techniques such as cubic utilization multiples to have more cargo space availability in shipping containers and trucks. With the help of proper cargo loading and routing optimization, firms can save cargo transportation costs per unit of volume and thus make overall supply chain better.

Factors and multiples are more than just abstract concepts used in mathematics; they are useful tools with real-world applications. From banking and engineering to science and education, understanding factors and multiples allows us to navigate and comprehend the complexities of our surroundings. By acknowledging their importance and investigating their applications, we can utilize the power of factors and multiples to promote innovation and advancement.

Related Articles

FAQs on Real-life Applications of Factors and Multiples

How are factors and multiples used in finance.

Factors and multiples determine interest rates, loan eligibility, and company performance metrics like P/E ratio. They’re also crucial in bond pricing.

How are factors and multiples applied in engineering?

Factors and multiples are used in structural design, electrical circuits, and manufacturing optimization.

What role do factors and multiples play in logistics and supply chain management?

Factors and multiples optimize inventory levels, route planning, and demand forecasting.

How do factors and multiples contribute to scientific research?

Factors and multiples are applied in biology (genetic traits), chemistry (stoichiometry), and physics (quantum mechanics).

What are some practical applications of factors and multiples in everyday life?

Factors and multiples are used in budgeting, time management, and cooking measurements.

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Problem-Solving Investigation: Multiples, factors & prime numbers (Year 6 Multiplication & Division)

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

15 April 2024

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Year 6 Multiplication and Division: Multiples, factors and prime numbers.

This in-depth Maths Investigation will develop maths meta-skills, and enable children to learn to think mathematically and articulate mathematical ideas.

In-depth Investigation: Magic Multiplication Squares Children complete a magic multiplication square using their knowledge of number properties and relationships. They then explore factors and multiples to create a new multiplication magic square.

This problem-solving investigation is part of our Year 6 Multiplication and Division block. Each Hamilton maths block contains a complete set of planning and resources to teach a terms worth of objectives for one of the National Curriculum for England’s maths areas.

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Year 6 Multiplication and Division - Problem-Solving Investigations

These in-depth maths investigations are open-ended problem solving activities for Year 6 children. **In-depth Investigation: Magic Multiplication Squares** Children complete a magic multiplication square using their knowledge of number properties and relationships. They then explore factors and multiples to create a new multiplication magic square. **In-depth Investigation: The Eights Have It** Children multiply numbers starting with 9 by 9 and add single-digit numbers in a decreasing sequence. They identify and describe the patterns and start to explain them. **In-depth Investigation: Awesome Answers** Using a magic square to generate 3-digit numbers, children create divisions with dividends containing specified fractions. **In-depth Investigation: Stunning Squares** Children explore patterns in the squares of numbers with reversed digits to find pairs of ‘stunning squares’. **In-depth Investigation: Geometry Genius** Children use what they know about how to find the areas of triangles and parallelograms to find the areas of rhombi, kites and trapezia. **In-depth Investigation: Get to the Root** Children use their fluency in mental multiplication to explore the patterns of digital roots in multiplication. **In-depth Investigation: Riveting Reversals** Multiply 3-digit numbers with consecutive digits by a 2-digit number; reverse the 3-digit number and repeat. Find the difference between the two answers. **In-depth Investigation: Why is it so?** Children identify a pattern in the division of a total of six numbers created using the same 3 digits. They then use algebra to explain why it is so. These investigations will develop maths meta-skills, support open-ended questioning and logical reasoning, and enable children to learn to think mathematically and articulate mathematical ideas. These problem-solving investigations come from our [Year 6 Maths Blocks](https://www.hamilton-trust.org.uk/maths/year-6-maths/). Each Hamilton maths block contains a complete set of planning and resources to teach a term’s worth of objectives for one of the National Curriculum for England’s maths areas.

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Multiples and Factors

The Ministry is migrating nzmaths content to Tāhurangi.             Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz).  When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024.  e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available. 

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This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework. A PDF of the student activity is included.

Click on the image to enlarge it. Click again to close. Download PDF (747 KB)

find factors and multiples of numbers

Number Framework Links Use this activity to develop knowledge about factors and multiples to complement advanced multiplicative part–whole strategies (stage 7).

FIO, Level 3, Multiplicative Thinking, Multiples and Factors, pages 16-17

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In this activity, students identify multiples and factors and solve problems that involve finding highest common factors and lowest common multiples. Students will need a good recall of multiplication basic facts in order to be able to do these activities.

Activity One and Game

Discuss the definitions of multiples and factors (on the student book page) before your students begin this activity. Make sure that they understand that every number is a factor of itself, because if they divide a number by itself, there is no remainder. For example, 12 ÷ 12 = 1 without a remainder, so 12 is a factor of 12. A prime number is a number that has only two factors, itself and 1, for example: 5, 7, 13, and 29. (Note that 1 itself is not considered to be a prime number.) Before the students play the game, ask the following questions: • Imagine you threw a 4 and a 6. Which squares could you choose to cover with your counter?(a number with more than two factors, a factor of 24, a multiple of 2, a multiple of 3, a multiple of 4, a multiple of 8, an even number, or a multiple of 6) • Imagine you need a multiple of 5 to get four counters in a row. Which throws of the dice would give you a multiple of 5? (1 and 5, 2 and 5, 3 and 5, 4 and 5, 5 and 5, 6 and 5, 7 and 5, 8 and 5, or 9 and 5)  

This game could be extended by asking:

• What are all the different products you could throw with the two game dice, one labelled 1–6 and the other 4–9? (4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 54)

• There are two different ways of getting a product of 12: throwing a 3 and a 4 or a 2 and a 6. Which other products can you throw more than one way using the game dice? (8: 1 x 8 or 2 x 4; 16: 2 x 8 or 4 x 4; 18: 2 x 9 or 3 x 6; 20: 4 x 5 or 5 x 4; 24: 3 x 8 or 4 x 6 or 6 x 4; 30: 5 x 6 or 6 x 5; 36: 4 x 9 or 6 x 6)

• What’s the probability of throwing a double? (There are 36 possible combinations that can be thrown with these dice, and only 3 of these are doubles: double 4, 5, or 6. So the probability of throwing a double is or .)

• Which squares in the game are easier/harder to cover? Can you use the information you have about the possible products that can be thrown to explain why? (Easier to cover: a number with more than two factors [34 out of 36 possible combinations have more than 2 factors; only 5 and 7 don’t], an even number, a multiple of 2 [27 out of 36 possible combinations are even and are therefore also multiples of 2], and a multiple of 3 [20 out of 36 possible combinations].

Harder to cover: a prime number [only 2 out of 36 combinations] and a multiple of 7 [only 6 chances out of 36].)

Activity Two

These problems ask students to find highest common factors and lowest common multiples. An understanding of these ideas is important for working with problems involving fractions and in algebra. It may help the students if they make a list of all the possible products that can be thrown with the two game dice so that they can then compare this list with the factors and multiples needed in the questions.

Answers to Activities

Activity One 1. 7, 14, 21, 28, 35, 42, 49, and so on 2. 1, 2, 3, 4, 6, 8, 12, 24 Game A game involving factors and multiples Activity Two 1. a. 8, because 8 is the highest factor that 16 and 24 have in common (highest common factor). The dice throws would be 1 and 8 or 2 and 4. b. 4 is the highest common factor of 12, 16, and 60. (The other common factors are 1 and 2.) 2. a. 21. (3 x 7 = 21) b. 30. (2 x 3 x 5 = 30. 15 is a multiple of 3 and 5 but not of 2, and the only even numbers less than 30 that are multiples of 5 are 10 and 20, neither of which is a multiple of 3.)

Numeracy Project materials (see Numeracy Books page )) • Book 8: Teaching Number Sense and Algebraic Thinking Prime Numbers (finding primes by representing numbers as rectangular arrays) Factor Trees (using factor trees to produce prime factors of a number) The Sieve of Eratosthenes (finding prime numbers) Highest Common Factors Lowest Common Multiples

Figure It Out • Basic Facts, Level 3 Factor Puzzles (using basic facts to identify factors) Stars and Students (game using multiplication facts to identify factors) • Basic Facts, Levels 3–4 Matrix (finding factors) A Matter of Factor (factor game) How Many Factors ? (factor investigation) Primates (finding prime factors)

nzmaths website A Prime Search (making arrays to explore prime numbers),

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