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Common Multiple: Definition and Example | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Common MultipleCommon Multiple: Definition and ExampleTable of ContentsDefinition of Common Multiples

Common multiples are numbers that are multiples of two or more given numbers. A multiple of a number is the product obtained when that number is multiplied by any counting number. For example, the multiples of 4 are 4, 8, 12, 16, and so on. When we identify numbers that appear in the multiple lists of two or more numbers, we call them common multiples. These shared multiples form an infinite set of numbers that are divisible by all the original numbers without leaving a remainder.

Common multiples can be found between two numbers or among three or more numbers. For two numbers, we simply list the multiples of each and identify the numbers that appear in both lists. For instance, the common multiples of 6 and 8 include 24 and 48. When working with three numbers, we need to find multiples that appear in all three lists. The smallest common multiple of any set of numbers is called the Least Common Multiple (LCM), which is particularly important in various mathematical operations, especially when working with fractions.

Examples of Common Multiples Example 1: Finding the multiples of 9 Problem:

What are the multiples of the number 9?

Step-by-step solution: Step 1, Recall that multiples of a number are found by multiplying it by consecutive natural numbers (1, 2, 3, etc.). Step 2, Let's create the list of multiples by multiplying 9 by each natural number: 9 ×\times× 1 = 9 9 ×\times× 2 = 18 9 ×\times× 3 = 27 9 ×\times× 4 = 36 9 ×\times× 5 = 45 Step 3, Think about the pattern: notice how each multiple increases by 9 from the previous multiple. Step 4, The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, and so on. This pattern continues infinitely. Example 2: Finding common multiples of 2 and 10 Problem:

Find two common multiples of the numbers 2 and 10.

Step-by-step solution:

Step 1, Let's list the multiples of 2:

2 ×\times× 1 = 2 2 ×\times× 2 = 4 2 ×\times× 3 = 6 2 ×\times× 4 = 8 2 ×\times× 5 = 10 2 ×\times× 6 = 12 2 ×\times× 7 = 14 2 ×\times× 8 = 16 2 ×\times× 9 = 18 2 ×\times× 10 = 20

So, the multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

Step 2, Let's list the multiples of 10:

10 ×\times× 1 = 10 10 ×\times× 2 = 20 10 ×\times× 3 = 30

So, the multiples of 10 are: 10, 20, 30, 40, 50, ...

Step 3, Examine both lists and identify the numbers that appear in both. Notice that 10 appears in both lists (it's the 5th multiple of 2 and the 1st multiple of 10). Similarly, 20 appears in both lists.

Step 4, Two common multiples of 2 and 10 are 10 and 20.

Example 3: Finding the LCM of 3 and 5 Problem:

Find the LCM of 3 and 5.

Step-by-step solution:

Step 1, Understand that the Least Common Multiple (LCM) is the smallest number that is divisible by both of the given numbers.

Step 2: List the multiples of 3:

3 ×\times× 1 = 3 3 ×\times× 2 = 6 3 ×\times× 3 = 9 3 ×\times× 4 = 12 3 ×\times× 5 = 15

So the multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...

Step 3: List the multiples of 5:

5 ×\times× 1 = 5 5 ×\times× 2 = 10 5 ×\times× 3 = 15

So the multiples of 5 are: 5, 10, 15, 20, 25, 30, ...

Step 4: Identify the common multiples by finding numbers that appear in both lists. Looking at our lists, we can see that 15, 30, ... are common to both.

Step 5: Since the LCM is the smallest common multiple, we identify 15 as the LCM of 3 and 5.

Step 6, When two numbers are co-prime (have no common factors except 1), their LCM equals their product. Since 3 and 5 are co-prime, their LCM is 3 ×\times× 5 = 15.

Step 7: The LCM of 3 and 5 is 15.

Comments(6)AActressPennyNovember 4, 2025I've used this glossary def for my students. It's super clear! Helped them grasp common multiples easily. Thanks!

NNatureLover85September 10, 2025I’ve been helping my kids with math, and the clear examples on this page made explaining common multiples so much easier! The step-by-step guide is a lifesaver for homework time.

NNatureLover85August 27, 2025I’ve used the Common Multiple definition and examples from this page to help my kids with their math homework. It’s super clear, and the step-by-step explanation makes it easy to follow!

NNatureLover92August 20, 2025I’ve been using this page to help my kids understand common multiples, and it’s been a game-changer! The examples are clear and the step-by-step approach made learning LCM so much easier for them.

MMathMom25August 6, 2025I’ve been using this definition to teach my kids about LCM, and it’s been a game-changer! The examples make it super easy for them to grasp the concept. Great resource!

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