LCM (Least Common Multiple)

The LCM stands for Lowest Common Multiple or Least Common Multiple. The smallest multiple that is common among two or more numbers is the LCM of the numbers. The least common multiple (LCM) is used to predict and schedule events. For example, here we take two values, 5 and 2.
Let's find the multiples of 2, which are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,...
Now, let's check for multiples of 5,
5, 10, 15, 20, 25, 30, 30, 40, 45,...
Here, 10 and 20 are common multiples of 2 and 5. The number 10 is the smallest. Thus, 10 is the least common multiple of 2 and 5.
 

LCM plays a crucial role in mathematics, as it is a fundamental concept in many branches of the subject. In this section, let’s learn about the importance of LCM in mathematics.
 

• Simplifying fractions with different denominators
• To predict or schedule recurring events 
• To simplify equations in algebraic problems 
   

There are various methods to find the LCM of any two numbers. The standard techniques include listing multiples, prime factorization, and division methods.

In this method, the multiples of the given numbers are listed to find the smallest common multiple. 


For example, 
LCM of 10 and 20
Multiples of 10 – 10, 20, 30, 40, …
Multiples of 20 – 20, 40, 60, 80, ….
Therefore, the LCM (10, 20) = 20.

LCM of 10, 20, and 30
Multiples of 10 – 10, 20, 30, 40, …
Multiples of 20 – 20, 40, 60, 80, ….
Multiples of 30 – 30, 60, 90, ……
Therefore, the LCM (10, 20, 30) = 60.
 

The product of the highest power of all the prime factors of the given numbers is the LCM.


For example, 
LCM of 10 and 20
Prime factorization of 10 = 2 × 5
Prime factorization of 20 = 2² × 5
LCM(10, 20) = 2² × 5 = 4 × 5 = 20.


LCM of 10, 20, and 30
Prime factorization of 10 = 2 × 5
Prime factorization of 20 = 2² × 5
Prime factorization of 30 = 2 × 3 × 5
LCM(10, 20) = 2² × 3 × 5 = 4 × 3 × 5 = 60.
 

In the division method, the given number is divided by its smallest common prime factor till we get 1. The LCM is the product of the divisors.

For example, 
LCM of 10 and 20
LCM (10, 20) = 2 × 2 × 5 = 20
LCM of 10, 20, and 30
LCM (10, 20, 30) = 2 × 2 × 3 × 5 = 60

\( \begin{array}{r|rr} 2 & 10 & 20 \\ \hline 2 & 5 & 10 \\ \hline 5 & 5 & 5 \\ \hline & 1 & 1 \end{array} \)


LCM of 10, 20, and 30
Prime factorization of 10 = 2 × 5
Prime factorization of 20 = 2² × 5
Prime factorization of 30 = 2 × 3 × 5
LCM(10, 20) = 2² × 3 × 5 = 4 × 3 × 5 = 60.


\( \begin{array}{r|rrr} 2 & 10 & 20 & 30 \\ \hline 2 & 5 & 10 & 15 \\ \hline 5 & 5 & 2 & 3 \\ \hline & 1 & 1 & 1 \end{array} \)

 

The concept of LCM is applied in our daily lives, such as event planning, scheduling buses, and more. Let us explore a few real-life applications of LCM. 
 

• LCM is used in daily life for various applications, including predicting and scheduling recurring events, among others. In this section, let’s learn about the real-world application of LCM. 
  
   
• Event planning: When the events recur in a fixed interval, we use LCM to predict the following events or when they co-occur.
  
   
• Buses running on schedules: To schedule the timing of buses, we use the LCM of the intervals to determine the time when the buses arrive.
  
   
• To synchronize events: When multiple processes occur at different intervals, we identify the standard time by finding the LCM.
   

As LCM is used in many branches of math, it is important to master LCM. In this section, let’s learn a few tips and tricks to master LCM. 

 

Understanding the concept of LCM:

Learning the basic concept of LCM makes it easy for students to master LCM. LCM is the smallest common multiple among two or more numbers.

 

Understanding the relationship between LCM and GCF:

The product of two numbers is the product of LCM and GCF. Therefore, LCM(a, b) = Product of a and b / GCF (a, b).

 

LCM of prime numbers: 
LCM of any prime number is the product of the prime numbers. For example, LCM of 5 and 11 is 5 ×11 = 55