LCM (Least Common Multiple)

The LCM, which stands for Lowest Common Multiple or Least Common Multiple, is the smallest multiple that is common among two or more numbers. In mathematics, the least common multiple (LCM) is used to predict and schedule events and plays an important role in arithmetic operations like finding a lowest common denominator when working with fractions.
 

To understand what is the least common multiple and how to find LCM, let’s take an example of how to do LCM for the numbers 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 the multiples of 5, which are 5, 10, 15, 20, 25, 30, 35, 40, 45, …

Here, 10 and 20 are common multiples of 2 and 5. Since 10 is the smallest multiple, it is the least common multiple (LCM) of 2 and 5. This is a simple least common multiple example.

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, 30) = 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, 30) = 2² × 3 × 5 = 4 × 3 × 5 = 60.


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

As LCM is used in many branches of math, it is important to master the concept. 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


Prime factorization method - Break each number into its prime factors and take all primes with the highest powers to find the LCM.


Using GCF to find LCM - Use the formula "LCM × GCF = Product of two numbers" to quickly calculate LCM if GCF is known.

LCM is used in our daily life for various purposes, including predicting and scheduling recurring events. In this section, let’s learn about the real-world applications 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.