How to Find Common Multiples of 2 and 3 with Easy Steps and Examples
When two numbers are multiplied together, the result is a multiple, or product. For instance, 20 is a multiple of 4 and 5 if we declare that 4 multiplied by 5 will give 20. The remaining multiples of four can be written as follows: 4, 8, 12, and so on as all of them are divisible by 4. Four times one equals four.
Common Multiples :
A whole number that is a shared multiple of every set of numbers is known as a common multiple. The term "common multiples" refers to the multiples that exist for two or more different numbers. On a grid of 100, let's mark the multiples of 6 and 7. The multiples of 6 will be marked with a circle, and the multiples of 7 with a cross.
Example of Common Multiples
Properties of Multiples :
The multiples' properties are listed below.
All numbers are multiples of one .
A number has an unlimited number of multiples.
A number's multiple is greater than or equal to the original number (except for 0).
How to Find Multiples of a Number:
When we multiply one whole number by another whole number, we have multiples as the result. Or, to put it simply, when you multiply it by a whole number, you receive its multiples.
Do you remember multiplication tables? They will be used by us to locate multiples. Let's identify the first five multiples of 6 that are not zero to see how that aids in our understanding of what multiples are.
Thus , Multiple of a number = Number Any integer (not a fraction).
Common Multiples of 2 and 3:
The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, ….
Hence, some of the common multiples of 2 and 3 are 6, 12, 18, etc.
Common Multiples of 3 and 5
The multiples of 3 are 3,6,9,12,15,18,21,24,27,30…..
The multiples of 5 are 5,10,15,20,25,30,35,40,45….
Hence, some common multiples of 3 and 5 are 15,30,45 etc
Conclusion :
A common multiple is a whole number that occurs multiple times in each set of numbers. A common multiple is a multiple that two or more numbers share.
Solved Examples:
1. List the first two multiples of 5 and 4 .
4, 8, 12, 16, 20, 24, 28, 32, 36, and 40,… are multiples of 4.
5, 10, 15, 20, 25, 30, 35, 40, and 45,... are multiples of 5.
20 and 40 are the two equivalent multiples of 5 and 4 respectively.
2. Find the least common multiples (LCM) of 3, 5 and 15.
First Let's list the multiples of 3, 5, and 15 that are common before determining the least common multiple.
Numerous multiples of 3 include 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and so forth.
5, 10, 15, 20, 25, 30, 35, 40, 45, and so on are examples of multiples of 5.
15, 30, 45, 60, 75, 90, 105, and other multiples of 15
15, 30,... and so on are common multiples of 3, 5, and 15 respectively. The smallest multiple among these regular multiples is 15. Therefore, the LCM of 3, 5, and 15 is 15 because it is the least of all the common multiples. Thus, 15 is the LCM of 3, 5 and 15.
3. List the first three multiples of 2 and 6 that are frequently repeated.
Numerous 2's include 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24.
Numerous 6 include: 6, 12, 18, 24, 30, 36, 42, and 48.
The first two common multiples of 2 and 6 are 6 and 12.
FAQs on Common Multiples of 2 and 3 Explained
1. What is the definition of a common multiple, with an example?
A common multiple is a whole number that is a shared multiple of two or more different numbers. In simpler terms, it's a number that you will find in the multiplication tables of all the numbers you are considering. For instance, to find the common multiples of 2 and 3, we can list their individual multiples:
- Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18...
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21...
2. How do you find the first three common multiples of 2 and 3?
To find the first three common multiples of 2 and 3, you can use the listing method. First, list the multiples of 2 and 3 separately until you find three matching numbers. The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, etc. The multiples of 3 are 3, 6, 9, 12, 15, 18, etc. By comparing these lists, you can identify the first three numbers that they share. Therefore, the first three common multiples of 2 and 3 are 6, 12, and 18.
3. What is the relationship between the common multiples of 2 and 3 and their Least Common Multiple (LCM)?
The Least Common Multiple (LCM) is the smallest of all the common multiples. For the numbers 2 and 3, the common multiples are 6, 12, 18, 24, and so on. The smallest number in this series is 6. Therefore, the LCM of 2 and 3 is 6. An important relationship to note is that all other common multiples of 2 and 3 are simply the multiples of their LCM (i.e., multiples of 6).
4. Why are all common multiples of 2 and 3 also multiples of 6?
This is a fundamental property related to the LCM. Since 6 is the Least Common Multiple (LCM) of 2 and 3, it is the first point where their multiplication patterns intersect. For a number to be a multiple of both 2 and 3, it must be divisible by both. Any number divisible by both 2 and 3 is, by definition, also divisible by their LCM, which is 6. This is why the sequence of common multiples (6, 12, 18, ...) is just the multiplication table of 6.
5. How are common multiples different from common factors for the numbers 2 and 3?
Common multiples and common factors are inverse concepts. It's a common point of confusion for students. Here is a clear comparison:
- Common Multiples are numbers you get by multiplying. They are always equal to or larger than the given numbers. The common multiples of 2 and 3 are 6, 12, 18, and so on.
- Common Factors are numbers you can divide by. They are always equal to or smaller than the given numbers. The only common factor of 2 and 3 is 1.
6. Can any two numbers have an infinite number of common multiples?
Yes, any given set of whole numbers will always have an infinite number of common multiples. Once you find the first common multiple (the LCM), you can simply keep finding multiples of that LCM, and this process can continue endlessly. For example, with 2 and 3, the common multiples 6, 12, 18, 24... form a sequence that never ends.
7. How is the concept of common multiples applied in real-life situations?
The concept of common multiples is essential for solving problems related to cycles, schedules, and events that need to align. Here are two real-world examples:
- Event Planning: If you are buying items that come in different pack sizes, like hot dog buns in packs of 8 and sausages in packs of 10, you would find the LCM (40) to determine the minimum number of each you need to buy to have no leftovers.
- Scheduling Tasks: If you water your flowers every 2 days and fertilise them every 3 days, the concept of common multiples tells you that you will do both tasks on the same day every 6 days (the LCM of 2 and 3).
8. How would you find the common multiples of three numbers, such as 2, 3, and 4?
The process is the same as with two numbers, but you must find a number that appears in the multiplication tables of all three.
- Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24...
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24...
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28...
