How to Find the Common Multiple of Numbers Easily
Multiples are used in whole mathematics. The extensive use of multiples is because of getting commons. It is used in calculation to simplify a fraction. The fundamental principle of arithmetic is also telling us about the multiples only.
Multiples
Multiples are numbers which we’ll get after multiplying a number with counting numbers. Let say we have a number n and we know that counting numbers are 1, 2, 3, 4, 5, 6…. Then, the product of n and 1, 2, 3, 4, 5, 6.. Is called multiples and this scenario is called multiplication.
Example: Let say we have a number 5. For getting it’s multiples we need to multiply it by counting numbers that are multiplication with 1, 2, 3, and so on. We’ll get
5 x 1 = 5
5 x 2 = 10
5 x 3 = 15
5 x 4 = 20
5 x 5 = 25
5 x 6 = 30
Hence, we’ll get multiples of 5 as 5, 10, 15, 20, 25 and so on.
Common Multiples
Whenever we want to talk about the common immediately, more than one quantity comes in our mind. Hence, we’ll start with 2 numbers whose multiples we have. Let say we have numbers as 5 and 6. First of all we’ll find there multiples.
Multiples of 5 will be 5, 10 ,15, 20, 25, 30, 35, 40, and so on.
Multiples of 6 will be 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 and so on.
Now to get common multiples we just need to find common numbers in both the multiples. Observe that in both the multiples set 30 is common. So, we’ll say 30 is the first common multiples or Least common multiple of 5 and 6. Least common multiple is commonly called LCM.
Now the obvious question comes in our mind, is there any multiple which is not least. The answer is yes. There are many common multiples after first or least. For example observe that for 5 and 6 first common multiples is 30 then second will be 60 third will be 90 and so on.
Representation of Common Multiples
Grid Method
Common multiples are the numbers which come common in both the multiples table of given numbers. If we take number 6 and 7 and we have to find the common multiples of them one way of representing them is following:
We have marked the circle for all the multiples of 6 and cross for all the multiples of 7 till 100 numbers. Now what will be our common multiples?
Common multiples will simply be those numbers that are crossed as well as circled. That’s how they’ll come in both the multiple tables.
Hence, 42, 84 and so on will be our common multiples. Also, the least common multiple will be 42.
Vann Diagram
Another way to represent common multiples is the Vann diagram. Suppose we have to find the common multiples of 3 and 4. Then we’ll close all the multiples of 3 and 4 in a circle separately. Then we’ll find commons in them. Kindly refer the following:
Solved Examples
1. Write the common multiples of 7 and 9.
Ans: First of all we’ll find the multiple of 7 which are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70. Now, multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 90. Now the Common multiples of 7 and 9 will be the number that occurs in both the tables. Which are 63, 126, 189, and so on.
2. A and B are two numbers such that B is a multiple of A. Which of the following is the value of LCM (A,B)?
Ans: We have given two numbers A and B. For finding the least common multiple first we need to find their common multiples. Then we can find least out of them. We have also given that B is a multiple of A means when we’ll be writing multiples of A then B will occur in its table. Also, all the common multiples will be the multiples of B. Hence the LCM(A, B) will be B.
Did You Know?
There are infinite common multiples possible for any two finite real numbers. We just have to write multiples of LCM.
The smallest common multiple is known as LCM. Which refers to least common multiple. The representation is LCM(a, b) which represents the least common multiple of a and b.
FAQs on Common Multiple: Meaning, Methods & Examples
1. What is a common multiple in mathematics?
A common multiple is a number that is a multiple of two or more different numbers. For example, to find the common multiples of 4 and 5, you list their individual multiples. Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40... and multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40... The numbers that appear in both lists, like 20 and 40, are their common multiples.
2. How do you find the first three common multiples of 3 and 6?
To find the first three common multiples of 3 and 6, you can use the listing method as per the NCERT syllabus for Class 4 to 6:
List the multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24...
List the multiples of 6: 6, 12, 18, 24, 30...
By comparing the two lists, you can identify that the first three numbers they share are 6, 12, and 18. These are the first three common multiples.
3. What is the difference between a common multiple and a common factor?
The key difference is their relationship with the original numbers. A common multiple is a number that two or more numbers can divide into without a remainder (e.g., 24 is a common multiple of 6 and 8). In contrast, a common factor is a number that can divide two or more numbers without a remainder (e.g., 2 is a common factor of 6 and 8). In essence, multiples are typically larger than the numbers, while factors are smaller or equal to them.
4. What are the main methods for finding the Least Common Multiple (LCM)?
The three primary methods for finding the LCM, which are part of the CBSE curriculum (2025-26), are:
Listing Method: Writing out the multiples of each number until you find the first one they have in common.
Prime Factorization Method: Finding the prime factors of each number and multiplying the highest power of every prime factor involved.
Common Division Method: Dividing the numbers together by their common prime factors and then multiplying all the divisors and the remaining numbers.
5. Why is finding the Least Common Multiple (LCM) more important than any other common multiple?
The LCM is especially important because it represents the most efficient common point. Its primary application is in adding and subtracting fractions with different denominators. To perform these operations, you need a common denominator, and the LCM provides the Least Common Denominator (LCD). Using the LCD simplifies the calculation significantly and helps avoid working with unnecessarily large numbers, which is a key problem-solving skill.
6. Can there be a 'Greatest Common Multiple' for two numbers? Explain why or why not.
No, two numbers cannot have a Greatest Common Multiple. The set of common multiples for any pair of numbers (except zero) is infinite. For instance, the common multiples of 5 and 10 are 10, 20, 30, 40, and so on, continuing without end. Since there is no largest number in this sequence, a 'greatest' common multiple doesn't exist. This is why mathematical focus is placed on the Least Common Multiple (LCM).
7. What is a real-world example of where common multiples are used?
A classic real-world example is planning recurring events. Imagine you water one plant every 4 days and another plant every 6 days. If you water both today, you can find out when you'll next water them on the same day by finding the Least Common Multiple (LCM) of 4 and 6. The LCM is 12, which means you will water both plants together again in 12 days.
8. What is the relationship between the HCF and LCM of two numbers?
For any two positive integers, 'a' and 'b', there is a fundamental relationship between their Highest Common Factor (HCF) and Least Common Multiple (LCM). The product of the HCF and LCM is always equal to the product of the two numbers themselves. This is expressed by the formula: HCF(a, b) × LCM(a, b) = a × b. This property is a useful shortcut for finding one value if you already know the other three.
