LCM stands for Least Common Multiple. It is a number theory topic. It is widely used in number theory and arithmetic. LCM value is useful in optimizing the quantity of the given objects. It is used to split things into smaller sections. It is also used to equally distribute any number of sets of items into their largest grouping. LCM has widely used in our daily life such as to figure out how many people we can invite. To arrange something into rows or groups. Also, LCM is very common in the life of students.
In this post, we will learn about the definition of LCM, the rules of LCM, and how to calculate LCM with a lot of examples we can also learn about GCF or HCF.
What is Common Multiple?
In order to learn about common multiple first, we have some sound knowledge about multiples.
Multiples of a number are those numbers that we get by multiplying that number by another number such as multiplying by 1, 2, 3, 4, etc., but not zero. Just like the multiplication table. e.g.,
The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ….
The multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, …
Now we come to our point of common multiples.
Common multiples are those multiples that are common in the multiples of two or more numbers. Let us take some examples to understand common multiples.
Example 1
Find the common multiples of 2 and 6.
Solution
Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …….
Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ….
Common Multiples of 2 and 6 = 6, 12, 18, …
Example 2
Find the common Multiples of 5 and 10.
Solution
Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ….
Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ….
Common Multiples of 5 and 10 = 10, 20, 30, 40, 50, …
What is LCM?
The LCM (least common multiple) of two or more numbers is the smallest non-zero common number which is a multiple of two or more numbers. When we take the common multiples, LCM is simply the smallest of the common multiples. It is used to split things into smaller sections. LCM is also known as LCD (least common divisor).
How to Find LCM?
We can calculate the LCM in three ways
- By List of Multiples
- By Prime factorization
- By Division method
All of the three methods to calculate LCM are used in the LCM calculator. This calculator will take the input and find the LCM with all three methods.
i. List of Multiples
In this method find all the multiples of two or more numbers then pick the common multiples and select the smallest one from the common multiples this smallest one is known as LCM. This method is applicable or quicker when you are dealing with smaller numbers.
Example 1
Find the common multiples of 2 and 6.
Solution
Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …….
Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ….
Common Multiples of 2 and 6 = 6, 12, 18, …
LCM of 2 and 6 = 6
As 6 is the smallest common multiple so 6 is the LCM of 2 and 6.
Example 2
Find the common Multiples of 5 and 10.
Solution
Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ….
Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ….
Common Multiples of 5 and 10 = 10, 20, 30, 40, 50, …
LCM of 5 and 10 = 10
As 10 is the smallest common multiple so 10 is the LCM of 5 and 10.
By Prime Factorization
Prime factors should be in prime numbers, prime numbers or coprime are those numbers that are only divisible by one and itself.
This method is applicable for larger numbers as a list of multiples of larger numbers is difficult. Prime factorization is a time-consuming method. Prime factorization follows these steps
- Find the prime factors of each number
- Circle common factors
- Write common factors and non-common factors
- Multiply common factors and non-common factors
The result is LCM. You can use the least common multiple calculator to ease up the process of calculating LCM using the prime factorization method.
Example 1
Find the least common Multiples of 42 and 28 by using prime factorization?
Solution
Prime factorization of 42 and 28 is.
| 2 | 42 |
| 3 | 21 |
| 7 | 7 |
| 1 |
| 2 | 28 |
| 2 | 14 |
| 7 | 7 |
| 1 |
Prime factors of 42 = 2x3x7
Prime factors of 28 = 2x2x7
Common factors of 42 and 28 = 2×7 = 14
Non common factors of 42 and 28 = 3×2 = 6
LCM of 42 and 28 = Common factors x non-Common factors
LCM of 42 and 28 = 14 x 6 = 84
Example 2
Find the least common Multiples of 60 and 90 by using prime factorization?
Solution
Prime factorization of 60 and 90 is.
| 2 | 60 |
| 2 | 30 |
| 3 | 15 |
| 5 | 5 |
| 1 |
| 2 | 90 |
| 3 | 45 |
| 3 | 15 |
| 5 | 5 |
| 1 |
Prime factors of 60 = 2x2x3x5
Prime factors of 90 = 2x3x3x5
Common factors of 60 and 90 = 2x3x5 = 30
Non common factors of 60 and 90 = 3×2 = 6
LCM of 60 and 90 = Common factors x non-Common factors
LCM of 60 and 90 = 30 x 6 = 180
Example 3
Find the least common Multiples of 112 and 168 by using prime factorization?
Solution
Prime factorization of 112 and 168 is.
| 2 | 112 |
| 2 | 56 |
| 2 | 28 |
| 2 | 14 |
| 7 | 7 |
| 1 |
| 2 | 168 |
| 2 | 84 |
| 2 | 42 |
| 3 | 21 |
| 7 | 7 |
| 1 |
Prime factors of 112 = 2x2x2x2x7
Prime factors of 168 = 2x2x2x3x7
Common factors of 112 and 168 = 2x2x2x7 = 56
Non common factors of 112 and 168 = 3×2 = 6
LCM of 112 and 168 = Common factors x non-Common factors
LCM of 112 and 168 = 56 x 6 = 336
By Division method
This method is simpler and time-consuming than the list of multiples and by prime factorization. In this method, we write two or more numbers in one box and then multiply the prime factors we got our output as LCM.
Example 1
Find the least common Multiples of 56 and 70 by using the division method?
Solution
Division of 56 and 70 are.
| 2 | 56, 70 |
| 2 | 28, 35 |
| 2 | 14, 35 |
| 5 | 7, 35 |
| 7 | 7, 7 |
| 1, 1 |
LCM of 56 and 70 = 2x2x2x5x7
LCM of 56 and 70 = 280
Example 2
Find the least common Multiples of 30, 40, and 60 by using the division method?
Solution
Division of 30, 40 and 60 is.
| 2 | 30, 40, 60 |
| 2 | 15, 20, 30 |
| 2 | 15, 10, 15 |
| 3 | 15, 5, 15 |
| 5 | 5, 5, 5 |
| 1, 1, 1 |
LCM of 30, 40 and 60 = 2x2x2x3x5
LCM of 30, 40 and 60 = 120
Summary
In this post, we have learned about the definition, rules, and examples of the least common multiple. Once you grabbed the knowledge of this topic you can easily solve the problems related to the least common multiple. It is not a complicated topic once you learn the basics of this topic you can solve any problem related to LCM.
