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Understanding Common Multiples in Mathematics
Common multiples refer to the multiples shared between a given set of numbers. When we multiply a number by counting numbers, we obtain its multiples. In mathematics, a multiple is the product or result of one number multiplied by another number. We often use multiplication tables to solve math problems, which involve finding multiples. The answer to the multiplication is the multiple of the two numbers.
For instance, let’s consider the multiples of 4:
- 4 times 1 = 4
- 4 times 2 = 8
- 4 times 3 = 12
- 4 times 4 = 16
- …
To better comprehend the concept of multiples, we can rely on multiplication tables representing the first ten multiples of numbers from 1 to 10. An example of a multiples table is displayed below:
Now, let’s explore the concept of common multiples through an example. Consider finding the common multiples of 6 and 7. We can list the multiples of 6 and 7 by multiplying them with the numbers 1, 2, 3, and so on.
Finding Common Multiples of 6 and 7
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 79, 84, …
The multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, …
The numbers common to both lists are: 42, 84, …
Defining Common Multiples
A common multiple is a number that multiple given set of numbers has in common. It refers to the multiples shared between two or more numbers.
Discovering Common Multiples Between Two Numbers
Finding common multiples between two numbers is simple. We list the multiples of both numbers, then identify the common multiples shared by both lists.
For example, let’s find the common multiples of 6 and 8:
The multiples of 6: 6, 12, 18, 24, 30, 36, …
The multiples of 8: 8, 16, 24, 32, 40, …
The common multiples of 6 and 8: 24, 48, …
Exploring Common Multiples Between Three Numbers
We can also find the common multiples of three numbers using the listing method. The key is to identify the numbers that are multiples of all three given numbers.
For example, let’s determine the common multiples of 3, 4, and 6:
The multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, …
The multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, …
The multiples of 6: 6, 12, 18, 24, 30, 36, …
The common multiples of 3, 4, and 6: 12, 24, …
How to Find Common Multiples
To find the common multiples of two or more numbers, we can list the multiples of each number.
Let’s consider finding the common multiples of 6 and 7. We can mark the multiples of 6 and 7 on a hundred grid to visualize them. Multiples of 6 will be marked with a circle, while multiples of 7 will be marked with a cross.
The common multiples of 6 and 7 are the numbers that are both circled and crossed on the grid. In this case, the common multiples are 42 and 84.
Let’s explore one more example:
Example: Find the common multiples of 4 and 12.
Multiples of 4: 4, 8, 12, 16, 20, 24, …
Multiples of 12: 12, 24, 36, 48, 60, 72, …
Common multiples of 4 and 12: 12, 24, …
Understanding the Least Common Multiple (LCM)
The smallest common multiple among two or more numbers is called the Least Common Multiple (LCM). To find the LCM, we list the multiples of the given numbers and determine their common multiples.
For instance, consider finding the LCM of 3 and 4:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, …
Multiples of 4: 4, 8, 12, 16, 20, 24, …
The common multiples of 3 and 4: 12, 24, …
Therefore, the smallest common multiple is 12, making the LCM of 3 and 4 equal to 12.
Properties of Common Multiples
- A number can have an infinite number of multiples, resulting in an infinite number of common multiples between any two numbers or sets of numbers.
- For any two numbers, multiplying them together will always yield a common multiple.
- If two numbers are coprime (having no common factors other than 1), their common multiples will be the multiples of their product.
- The LCM of two coprime numbers is equal to the product of the two numbers.
- If one number is a multiple of another, the LCM of the two numbers will be the multiple.
These properties assist in solving problems related to common multiples.
Practice Problems on Common Multiples
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What are the multiples of the number 9?
Solution:
The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, … -
Find two common multiples of the numbers 2 and 10.
Solution:
The multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
The multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, …
The two common multiples of 2 and 10 are 10 and 20. -
Find the LCM of 3 and 5.
Solution:
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, 50, …
The common multiples of 3 and 5 are: 15, 30, …
The smallest common multiple is 15, so the LCM of 3 and 5 is 15.
For more practice problems on common multiples, visit 5 WS.
Frequently Asked Questions About Common Multiples
Q: What are the multiples of any given number, such as 9?
A: The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …
Q: Can you provide the common multiples of 2 and 10?
A: The common multiples of 2 and 10 are 10 and 20.
Q: How do you find the LCM of two numbers, like 3 and 5?
A: To find the LCM of 3 and 5, we determine their common multiples, which in this case are 15, 30, … The smallest common multiple is 15, so the LCM of 3 and 5 is 15.
These are some frequently asked questions regarding common multiples.
