What are Common Multiples in Math?
Common multiples are the multiples that are common between a given set of numbers.
Multiplying a number by counting numbers gives us its multiples. In math, the meaning of a multiple is the product or result of one number multiplied by another number.
We all know multiplication tables as we always use them to solve math problems. When we use these tables, we are also using multiples. You see, when we multiply two numbers, the answer is their multiple.
Take a look at the multiplication tables of numbers from 1 to 10. Here, we obtained multiplies of these numbers by multiplying them with natural numbers.
Alt Tag: Multiplication facts of 1 to 10
So, what are common multiples? Let us try and understand with an example. It is quite simple. Let’s find the common multiples of 6 and 7.
First, we can list the multiples of 6 and 7 by multiplying them by numbers 1, 2, 3, …, and so on.
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, …
So, what numbers do you find common in the multiples of 6 and 7?
Common multiples of 6 and 7: 42, 84, …
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Common Multiples Definition
A common multiple is defined as a whole number which is a shared multiple of a given set of numbers. The multiples common to two or more numbers are called the common multiples of those numbers.
How To Find Common Multiples
We can find the common multiples of two or more numbers by listing the multiples of each number.
Consider the previous example. To find the common multiples of 6 and 7, let us mark the multiples of 6 and 7 on a hundred grid. We will mark the multiples of 6 by a circle and multiples of 7 by a cross.
The numbers that are circled as well as crossed are the common multiples of 6 and 7.
So, the common multiples of 6 and 7 are 42 and 84.
Take a look at one more example.
Example: Find common multiples of 4 and 12.
Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, …
Multiples of 12 = 12, 24, 36, 48, 60, 72, …
Common multiples of 4 and 12 = 12, 24, …
What is a Least Common Multiple?
The smallest common multiple of two or more numbers is called the Least Common Multiple (LCM).
For example, to find the common multiples of 3 and 4, we list their multiples and then find their common multiples.
The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
The common multiples of 3 and 4 are: 12, 24, 36, ….
The smallest common multiple is 12. So the LCM of 3 and 4 is 12.
Properties of Common Multiples
- A number can have an infinite number of multiples. Therefore, any two numbers or set of numbers can have an infinite number of common multiples.
- For any two numbers a and b, the product $a \times b$ is always a common multiple of a and b.
Example: $6 \times 8 = 48$ is a common multiple of 6 and 8.
- If two numbers a and b are coprime, then their common multiples are the multiples of $(a \times b)$.
Example: 2 and 3 are coprime numbers.
$2 \times 3 = 6$
Common multiples of 2 and $3 = 6,\; 12,\; 18,\; 24,\; 30$, …
- The LCM of two coprime numbers is equal to the product of the two numbers.
- If b is the multiple of a, then the LCM of a and b is b.
Example: LCM $(5,\; 10) = 10$
Solved Examples on Common Multiples
1. What are the multiples of the number 9?
Solution:
We know that we can get multiples of a number by multiplying it by 1, 2, 3, …, and so on.
So, the multiples of 9 are : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …
2. Find two common multiples of the numbers 2 and 10.
Solution:
We know that the multiples common to two or more numbers are called the common multiples of those numbers.
Now, the multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
and the multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, …
So, the two common multiples of 2 and 10 are 10 and 20.
3. Find the LCM of 3 and 5.
Solution:
The smallest common multiple of two or more numbers is called the Least Common Multiple (LCM).
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 of these is 15. So the LCM of 3 and 5 is 15.
Practice Problems On Common Multiples
Common Multiples - Definition with Examples
Which of the following is not a multiple of 8?
The multiples of 8 are: 8, 16, 24, 32, 40, …
So, 28 is not a multiple of 8.
Which of the following is a common multiple of 16 and 20?
Now,the multiples of 16 are $\colon$16, 32, 48, 64, $\underline{80}$, 96, 112, 128,144, $\underline{160}\dots$
and the multiples of 20 are $\colon$20, 40, $\underline{60}$, 80, 100, 120, 140, $\underline{160}$, 180, 200$\dots$
So, the common multiples of 16 and 20 are $\colon$80 and 160, of which 80 is option d.
Which of the following is not a common multiple of 12 and 15?
30 is a multiple of 15, but not of 12. So, 30 is not a common multiple of 12 and 15.
Which of the following is the LCM of 9 and 18?
18 is the smallest common multiple of 9 and 18. So, 18 is the LCM of 9 and 18.
Frequently Asked Questions On Common Multiples
What are the multiples of the number zero?
The multiples of zero is zero. Every other whole number has infinitely many multiples.
For example: 25 × 0 = 0 ; 1.0836 × 0 = 0 ; -9/87 × 0 = 0.
What are the multiples of the number one?
All the natural numbers are multiples of 1. There is no end to multiples of any number. The first ten multiples of the number 1 starting from 1 are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
What is the LCM of prime numbers?
The short form LCM stands for Least Common Multiple. The smallest common multiple of two or more prime numbers is their product.
For example: The LCM of 3 and 7 is 21.