Introduction
In mathematics, a multiple is a number that is the product of a given number and any other whole number. For example, the multiples of 3 are 3, 6, 9, 12, and so on. In this article, we will focus on common multiples, which are multiples that are shared by two or more numbers.
A common multiple of two or more numbers is a number that is a multiple of each of them. For example, the common multiples of 2 and 3 are 6, 12, 18, 24, and so on. These are all multiples of 2 and 3. Common multiples are important in mathematics because they help us to find the least common multiple, which is the smallest number that is a multiple of two or more numbers.
To understand common multiples, it’s important to first understand what multiples are. A multiple of a number is any number that can be obtained by multiplying the original number by an integer (a whole number). For example, the multiples of 5 are 5, 10, 15, 20, 25, and so on. Each of these numbers can be obtained by multiplying 5 by an integer, such as 1, 2, 3, 4, and so on.
When we talk about common multiples, we’re referring to the multiples that two or more numbers have in common. For example, let’s consider the numbers 6 and 9. The multiples of 6 are 6, 12, 18, 24, 30, and so on, while the multiples of 9 are 9, 18, 27, 36, and so on. As you can see, the number 18 is a multiple of both 6 and 9, so it’s a common multiple of these two numbers.
We can find common multiples by listing the multiples of each number and then looking for the ones they have in common. This process can be tedious, especially when dealing with larger numbers. For example, if we want to find the common multiples of 3 and 4, we might start by listing their 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, …
To find the common multiples, we need to look for the numbers that appear in both lists. In this case, the common multiples are 12, 24, and so on. We could continue listing the multiples, but it’s clear that any number that is a multiple of both 3 and 4 will be a common multiple.
While listing the multiples of each number is one way to find common multiples, there’s a more efficient method that involves finding the least common multiple (LCM) of the two numbers. The LCM is the smallest number that is a multiple of both numbers. To find the LCM of two numbers, we can use the prime factorization method.
The prime factorization of a number is the product of its prime factors. For example, the prime factorization of 24 is 2 × 2 × 2 × 3, since 24 can be written as 2 × 2 × 2 × 3. To find the LCM of two numbers, we can first find their prime factorizations and then take the product of the highest power of each prime factor.
Definition of Common Multiples
A common multiple is a multiple that is shared by two or more numbers. For example, the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, and so on. The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, and so on. The common multiples of 3 and 4 are 12, 24, 36, 48, and so on. Note that 12, 24, and 48 are common multiples because they are multiples of both 3 and 4.
In general, if we have two or more numbers, we can find their common multiples by finding the multiples of each number and then finding the numbers that are common to all of them.
Examples of Common Multiples
Example 1: Find the common multiples of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, and so on.
The multiples of 6 are 6, 12, 18, 24, 30, 36, and so on.
The common multiples of 4 and 6 are 12, 24, and so on.
Example 2: Find the common multiples of 7, 8, and 9.
The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, and so on.
The multiples of 8 are 8, 16, 24, 32, 40, 48, and so on.
The multiples of 9 are 9, 18, 27, 36, 45, and so on.
The common multiples of 7, 8, and 9 are 504, 1008, and so on.
Example 3: Find the common multiples of 2, 3, and 4.
The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, and so on.
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, and so on.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, and so on.
The common multiples of 2, 3, and 4 are 12, 24, and so on.
Example 4: Find the common multiples of 5 and 7.
The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, and so on.
The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, and so on.
The common multiples of 5 and 7 are 35, 70, and so on.
Quiz
What is a common multiple of 3 and 5?
What is the smallest common multiple of 6 and 9?
What is the LCM of 12 and 16?
What is a common multiple of 4, 6, and 9?
What is the smallest common multiple of 8 and 10?
What is the LCM of 18 and 24?
What is a common multiple of 7 and 8?
What is the LCM of 15 and 25?
What is a common multiple of 10 and 12?
What is the LCM of 20 and 30?
