102 LearnersLast updated on July 4th, 2025
Common Multiples
Common multiples are multiples shared by a given group of numbers. For example, 12 is a common multiple of 2 and 3; 2 6 = 12, and 3 4 = 12. To find the common multiples of two or more numbers, we should list the multiples of the given numbers separately and then identify the common numbers among the multiples.
What Are Common Multiples?
When we multiply a number by another number, the result that we get is called a multiple. For example, the multiples of 5 are 5, 10, 15, 20, 25, 30,...and so on. Here, the number 5 is multiplied by 1, 2, 3, 4, 5, 6, … and so on. The multiples of two or more numbers are called common multiples.
For example, the multiples of 2 are 2, 4, 6, 8, 10, 12,..., and the multiples of 3 are 3, 6, 9, 12, 15, 18,...therefore, 6, 12, 18, 24, 30,…are the common multiples of 2 and 3. The smallest common multiple is called the Least Common Multiple (LCM), which in this case is 6.
Difference Between Factors and Multiples
A multiple of a number is the product of that number and any whole number. For example, the multiples of 5 are 5, 10, 15, 20, 25, 30, etc. On the other hand, a factor is a whole number that divides another number evenly. For example, the factors of 5 are 1 and 5; 5 is a prime number and the factors of a prime number are always 1 and the number itself.
Let’s understand this better using a table
|
Factors |
Multiples |
|
Factors are whole numbers that divide another number evenly without a remainder. |
Multiples are numbers you get when you multiply a number by other whole numbers. |
|
Factors are smaller than or equal to the number. |
Multiples are greater than or equal to the number. |
|
The number of factors is limited. |
The number of multiples is unlimited. |
|
Example: Factors of 6 are 1, 2, 3, and 6. |
Example: Multiples of 6 are 6, 12, 18, 24, 30... |
How to Find the Multiples of a Number?
The multiples of a given number can be found by multiplying the number with natural numbers like 1, 2, 3, 4, 5, and so on. For example, if you want to find the multiples of 4, multiply 4 by 1, 2, 3, 4, 5, 6, etc.
4 x 1 = 4
4 x 2 = 8
4 x 3 = 12
4 x 4 = 16
4 x 5 = 20
4 x 6 = 24
So, the multiples of 4 are 4, 8, 12, 16, 20, 24,... and it goes on forever.
We can also find the multiples of a number by repeatedly adding the number to itself. For example, to find the multiples of 10, we keep adding 10 each time. The multiples we get are 10, 20, 30, 40, etc.
Common Multiples of Two Numbers
To find the common multiples of two numbers, we first list the multiples of each number. Then, we look for the numbers that appear in both lists, called common multiples. For example, let’s take 3 and 4 and common multiples in the first 10 multiples.
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40.
The numbers 12, and 24 are the common multiples of 3 and 4.
Common Multiples of Three Numbers
Just like a common multiple of two, to find the common multiples of three numbers, we first list the multiples of each number. Then, we identify the numbers that are present in all three lists. For example, let’s consider the first 10 multiples of 2, 3, and 4 and find the common multiples among them.
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
The number that is common to all three numbers is 12. This is the only common multiple of 2, 3, and 4 in the first 10 multiples of each number.
Least Common Multiple
LCM is the smallest multiple that can be divided evenly by two or more numbers. There are different methods to calculate LCM like the listing method, prime factorization, and division. For example, let’s find the LCM of 4 and 6:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28,...
Multiples of 6: 6, 12, 18, 24, 30, 36,...
The common multiples of 4 and 6 are 12, 24, 36, and so on. The smallest common multiple is 12. Therefore, the LCM of 4 and 6 is 12, as it is the smallest number that is divisible by both 4 and 6.
Let’s look at the multiples of the first 10 numbers in a table.
|
Number |
Multiples |
| 1 |
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... |
| 2 |
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... |
| 3 |
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... |
| 4 |
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ... |
| 5 |
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ... |
| 6 |
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ... |
| 7 |
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, ... |
| 8 | 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, ... |
| 9 |
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ... |
| 10 |
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ... |
Real-Life Applications of Common Multiples
Common multiples play an important role in solving real-life problems where events or actions repeat at regular intervals. Here are some everyday situations where common multiples are used.
- Traffic lights timing: Traffic lights at different intersections change after fixed intervals. To make them work smoothly, the time intervals are set using the Least Common Multiple
- School Timetable: If a music class happens every 3 days and a sports class every 5 days, the LCM helps find when both classes will happen on the same day.
- Event Planning: When planning events, LCM is used to find when two or more activities will occur together, like scheduling buses or cleaning schedules.
Common Mistakes and How to Avoid Them in Common Multiples
When finding common multiples, students often make mistakes that can lead to incorrect answers. Here are some common errors and tips to avoid them.
Mistake 1
Writing only a few multiples and missing the common ones
Students might miss writing some of the common multiples while calculating or writing the final answer. Always recheck the common multiples before writing them down.
Mistake 2
Forgetting to find the smallest common multiple (LCM)
After listing the common multiples, always check for the smallest one, as that is the LCM.
Mistake 3
Skipping numbers while listing multiples
Use proper multiplication tables and carefully write each multiple without missing any number.
Mistake 4
Confusing multiples with factors
Remember that multiples are products of numbers, while factors are numbers that divide exactly.
Mistake 5
Only comparing two numbers when more are given
List the multiples of all the given numbers separately to avoid confusion. Once the multiples are listed, find the common ones.
Solved Examples for Common Multiples
Problem 1
What are the common multiples of 3 and 4 up to 30?
The common multiples are 12 and 24.
Explanation
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30..., and the multiples of 4 are 4, 8, 12, 16, 20, 24, 28... The numbers 12 and 24 appear in both lists, so they are common multiples.
Problem 2
What is the smallest common multiple of 8 and 12?
The smallest common multiple is 24.
Explanation
The multiples of 8 are 8, 16, 24, 32..., and the multiples of 12 are 12, 24, 36.... The first common multiple is 24, which is the Least Common Multiple (LCM).
Problem 3
John is stacking boxes in piles of 8 and 12. What is the smallest number of boxes that will be the same in both piles?
The smallest number of boxes is 24.
Explanation
The multiples of 8 are 8, 16, 24, 32, etc., and the multiples of 12 are 12, 24, 36, etc. The smallest common multiple is 24.
Problem 4
Find the common multiples of 5 and 6 up to 50.
The common multiple up to 50 is 30. Others may appear beyond that.
Explanation
The multiples of 5 up to 50 are 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. The multiples of 6 up to 50 are 6, 12, 18, 24, 30, 36, 42, and 48. The common multiple is 30.
Problem 5
A train stops at stations every 10 minutes, and a bus stops every 15 minutes. How often do they stop at the station together?
They stop together every 30 minutes
Explanation
The multiples of 10 are 10, 20, 30, etc., and the multiples of 15 are 15, 30, 45, etc. The smallest common multiple is 30.
FAQs Related to Common Multiples
1.What is a common multiple?
2.What is the smallest common multiple called?
3.Can a number have unlimited common multiples?
4.How is the LCM different from the greatest common factor (GCF)?
5.Can the LCM of two numbers ever be smaller than the given numbers?
6.How can children in Canada use numbers in everyday life to understand Common Multiples ?
7.What are some fun ways kids in Canada can practice Common Multiples with numbers?
8.What role do numbers and Common Multiples play in helping children in Canada develop problem-solving skills?
9.How can families in Canada create number-rich environments to improve Common Multiples skills?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
