Factors and Multiples
Factors and multiples are foundational concepts in mathematics. They help us understand the relationship between numbers and how they can be broken down or combined. From dividing sweets equally among friends to solving puzzles or organising events, these concepts play an important role in our daily lives.
This guide will help you explore factors and multiples in detail, with step-by-step explanations, examples, and fun activities.
Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
What Are Factors?
A factor is a number that divides another number exactly without leaving any remainder. For instance, we know that 4 × 9 = 36. So both 4 and 9 can divide 36 exactly without remainder. And so both 4 and 9 are factors of 36.
In simple terms, factors are the “building blocks” of a number. They show how a number can be divided into smaller, equal parts.
Example: Factors of 36
While we know that 4 × 9 = 36 and so both 4 and 9 are factors of 36, there are other combinations that result in a product of 36.
For example, 6 × 6 = 36. Hence to find all the factors of 36, we must check systemically which numbers divide 36 exactly:
• 1 × 36 = 36 (both 1 and 36 are factors)
• 2 × 18 = 36 (both 2 and 18 are factors)
• 3 × 12 = 36 (both 3 and 12 are factors)
• 4 × 9 = 36 (both 4 and 9 are factors)
• 6 × 6 = 36 (6 is a factor)
Note that after 6 × 6, we will start to have repeating lines such as 9 × 4 = 36 and 12 × 3 = 36. But we do not need to write them down since those factors have already been listed.
Based on the systematic listing above, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Each of these numbers divides 36 exactly without leaving any remainder.
Why Are Factors Useful?
Factors are important in various areas of maths. For example:
They help in dividing things into equal groups. For instance, if you have 36 apples and want to share them equally among 9 friends, you need to know that 9 is a factor of 36.
Factors are also useful for simplifying fractions, such as reducing 12/36=1/3, since 12 and 36 both share the factor 12.
How to Find Factors
To determine the factors of a number:
- Start with 1 and the number itself. These are always factors.
- Check all smaller numbers to see if they divide the given number without leaving a remainder.
- List all the numbers that meet this condition. Each of these numbers is a factor.
- Stop listing when there are no other combinations between two listed factors on the left of the equal (=) sign. For example, in the process of listing factors of 30, we will obtain the following:
- 1 × 30 = 30
- 2 × 15 = 30
- 5 × 6 = 30
Observe that the numbers in the column on the far left will increase and the numbers in the centre column will decrease. When you observe that there are no more numeric combinations between 5 and 6, you can stop listing.
Practice Questions for Factors
1. Find all the factors of 35.
Solution [systematic listing]:
- 1 × 35 = 35
- 5 × 7 = 35 [See that there are no more combinations between 5 and 7, so the listing stops here]
The factors of 35 are 1, 5, 7 and 35.
2. Find all the factors of 48.
Solution [systematic listing]:
- 1 × 48 = 48
- 2 × 24 = 48
- 3 × 16 = 48
- 4 × 12 = 48
- 6 × 8 = 48 [See that there are no more combinations between 6 and 8, so the listing stops here]
What Are Multiples?
A multiple is a number that results when you multiply a given number by a whole number.
Multiples can be thought of as “stretching” the number because the number increases as the multiplier grows.
Example: Multiples of 5
Recall the 5 times table. To find the multiples of 5, we multiply it systematically by different whole numbers:
- 5 × 1 = 5
- 5 × 2 = 10
- 5 × 3 = 15
- 5 × 4 = 20
- 5 × 5 = 25
Thus, the multiples of 5 are 5, 10, 15, 20, 25, and so on.
Why Are Multiples Useful?
Multiples are essential for understanding patterns and solving problems such as:
- Finding common multiples of two or more numbers.
- Determining when two repeating events (like buses arriving) will occur at the same time.
- Creating mathematical sequences, such as 5, 10, 15, 20, …
How to Find Multiples
- Start with the number itself as the first multiple.
- Multiply the number by 1, 2, 3, and so on to find the next multiples.
- Continue multiplying to generate as many multiples as needed.
Practice Question for Multiples
1. Write down the first six multiples of 6 and 8. Identify the smallest number that appears in both lists.
- 6 × 1 = 6 8 × 1 = 8
- 6 × 2 = 12 8 × 2 = 16
- 6 × 3 = 18 8 × 3 = 24
- 6 × 4 = 24 8 × 4 = 32
- 6 × 5 = 30 8 × 5 = 40
- 6 × 6 = 36 8 × 6 = 48
The first six multiples of 6 are 6, 12, 18, 24, 30 and 36.
The first six multiples of 8 are 8, 16, 24, 32, 40 and 48.
The smallest number that appears in both lists is 24.
Key Differences Between Factors and Multiples
Factors and multiples are related concepts, but they have distinct differences. Understanding these differences will help you solve problems more easily.
| Factors | Multiples |
| Factors are smaller than or equal to the number. | Multiples are equal to or larger than the number. |
| A number has a limited number of factors. | A number has an unlimited number of multiples. |
| Factors divide a number exactly. Think of factors as “building blocks” to make larger numbers. Eg, 3 × 4 = 12, then 3 and 4 are factors of 12. | Multiples are products of the number. Associate multiples to the numbers in the multiplication table. Eg, 3, 6, 9 ,12 , … are multiples of 3. |
Common Factors and Highest Common Factor (HCF)
Common Factors
When two or more numbers share the same factors, these are called common factors.
Example: Common Factors of 24 and 36
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Common factors: 1, 2, 3, 4, 6, and 12
Highest Common Factor (HCF)
The HCF is the largest factor that two or more numbers share. In the example above, the HCF of 24 and 36 is 12. Because factors divide a number exactly without remainder, the HCF between a group of numbers is also called the Greatest Common Divisor (GCD).
Practice Question for Highest Common Factor
1. Find the HCF of 18 and 27.
Solution:
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 27: 1, 3, 9, 27
HCF of 18 and 27 is 9.
Common Multiples and Lowest Common Multiple (LCM)
Common Multiples
Common multiples are multiples that two or more numbers share.
Example: Common Multiples of 3 and 5
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21…
- Multiples of 5: 5, 10, 15, 20, 25…
- Common multiples: 15, 30, 45…
Lowest Common Multiple (LCM)
The LCM is the smallest multiple that two or more numbers share. In the example above, the LCM of 3 and 5 is 15.
Practice Question for Lowest Common Multiple
1. Find the LCM of 8 and 14.
Solution:
- Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, …
- Multiples of 14: 14, 28, 42, 56, …
LCM of 8 and 14 is 56.
Common Problem Sums Involving HCF and LCM
1. Sharing Equally
Mdm Siti bought 60 highlighters, 72 markers and 90 pencils to be packed into some goodie bags.
(a) What is the largest number of goodie bags that can be packed so that the items in each goodie bag are the same?
(b) List the items in each goodie bag.
Solution:
(a)
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
HCF of 60, 72 and 90 is 6.
Ans: The largest number of goodie bags is 6.
(b)
60 = 6 × 10
72 = 6 × 12
90 = 6 × 15
Each goodie bag will contain 10 highlighters, 12 markers and 15 pencils.
2. Tiling
A rectangular piece of paper measures 84 cm by 60 cm. The paper will be cut into squares of equal lengths while ensuring no paper is leftover. What is the least number of squares that can be cut?
Solution:
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
HCF of 84 and 60 is 12. Each square will measure 12 cm by 12 cm.
84 = 12 × 7
60 = 12 × 5
Along the length of 84 cm, 7 squares can be cut. And along the breadth of 60 cm, 5 squares can be cut. So the total least number of squares obtained is
7 × 5 = 35.
3. Events Occurring At The Same Time
At a bus and train interchange station, a bus arrives every 12 minutes and a train arrives every 15 minutes. The last time a bus and a train arrive together was 2 pm. How many times will a bus and a train arrive at the same time from 2:30 pm to 6:00 pm?
Solution:
- Multiples of 12: 12, 24, 36, 48, 60, 72, …
- Multiples of 15: 15, 30, 45, 60, …
The LCM of 12 and 15 is 60.
The bus and train will arrive at the same time every 60 min. The times both bus and train arrive after 2 pm will be 3pm, 4pm, 5pm and 6pm.
Ans: 4
Prime Numbers and Composite Numbers [Only Tested in Secondary Syllabus]
Prime Numbers
A prime number is a number that has exactly two factors: 1 and itself. This means that prime numbers cannot be divided by any number other than these two.
Examples
The first few prime numbers are:
- 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
Note that the number 2 is the smallest prime and the only even prime number. All other even numbers are composite [more on this in the next section] because they have at least three factors.
Composite Numbers
A composite number has more than two factors. For example: 12 is composite because its factors are 1, 2, 3, 4, 6, and 12.
A number that is not prime, must be composite (with the exception of the number 1)
Special Case: The Number 1
The number 1 is unique because it has only one factor (being itself only). Therefore, it is neither prime nor composite.
Practice Questions for Prime Numbers and Composite Numbers
1. Is 37 a prime number? Explain your answer.
Solution:
Yes. 37 has exactly two factors (1 and 37) and so 37 is a prime number.
2. Is 99 a composite or prime number? Explain your answer.
Solution:
1 × 99 = 99
3 × 33 = 99
Since 99 has more than 2 factors (1, 3, 33, 99, …) , 99 is a composite number.
Note that you do not have to list all the factors of 99 to show 99 is not prime. You only need to list more than 2 factors.
Learning Factors and Multiples
Understanding factors and multiples is a crucial step in mastering mathematics. These concepts not only help in solving problems but also build a strong foundation for advanced topics. Practise regularly, explore real-life examples, and enjoy the journey of learning maths!
Need More Revision Notes for PSLE Math?
Check out our exam guide on other topics here!
Before you go, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox.
Does your child need help with Mathematics?
1) Live Zoom Lessons at Grade Solution Learning Centre
At Grade Solution Learning Centre, we are a team of dedicated educators whose mission is to guide your child to academic success. Here are the services we provide:
– Live Zoom lessons
– Adaptably, an AI-Powered learning platform that tracks your child’s progress, strengths and weaknesses through personalised digital questions.
– 24/7 Homework Helper Service
We provide all these services above at a very affordable monthly fee to allow as many students as possible to access such learning opportunities.
We specialise in English, Math, Science, and Chinese subjects.
You can see our fees and schedules here >>
2) Pre-recorded Online courses on Jimmymaths.com
If you are looking for something that fits your budget, or prefer your child learn at his or her own pace, you can join our pre-recorded online Math courses.
Your child can:
– Learn from recorded videos
– Get access to lots of common exam questions to ensure sufficient practice
– Get unlimited support and homework help
You can see the available courses here >>