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Last updated on September 29, 2025

LCM (Least Common Multiple)

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Do you need a number that two different numbers can both divide into evenly? Then use the Least Common Multiple (LCM) to find the number.

LCM (Least Common Multiple) for US Students
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What is 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.
 

To make you understand better, let's find the LCM of 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. Since 10 is the smallest multiple, it is the least common multiple of 2 and 5.
 

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Importance of LCM in Mathematics

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 
     
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Methods to Find LCM

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

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Method 1: Listing the Multiples

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.
 

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Method 2: Prime Factorization Method

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.
 

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Method 3: Division Method

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.


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

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Tips and Tricks to Master LCM

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.

 

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Common Mistakes and How to Avoid Them in LCM

LCM is used in different branches of math, and students tend to make errors when finding LCM. Mistakes are common among students and in this section, we will explore common mistakes.

Mistake 1

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Misunderstanding the concept of LCM and GCF

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Confusing LCM and GCF is a common error made by students. To avoid it, students should be aware of the concept of LCM and GCF. LCM is about multiples and GCF is about the factors.

Mistake 2

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Errors in the division method

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While using the division method, the number should be divided only by the prime factors. And the process should be continued until all the numbers become 1. 

Mistake 3

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Confusion with the relationship between LCM and GCF

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Students tend to be confused with the relation between the LCM and GCF. The relationship between LCM and GCF is that the product of the number is equal to the product of LCM and GCF. 

Mistake 4

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Assuming that the product of two numbers is the LCM

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Students think that the product of two numbers is the LCM, but it is not true in all cases. It only works for co-primes. 

Mistake 5

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Errors in listing multiples

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When listing multiples to find the LCM, students may list multiples incorrectly. So try to double-check the multiples and verify whether it's correct or not.

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Real-World Applications of LCM

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.
     
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Solved Examples of LCM

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Problem 1

If a bus and train come at an interval of every 5 and 10 minutes respectively, find the time when they both come together.

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Both the bus and the train come together once in every 10 minutes.

Explanation

To predict the event, we find the LCM of the intervals.


Here we find the LCM of 5 and 10


Multiples of 5 are 5, 10, 15, 20, 30,…


Multiples of 10 are 10, 20, 30,….


LCM (5, 10) = 10.

 

Therefore, the answer is once in every 10 minutes.
 

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Problem 2

Sam has swimming classes every 5 days and drawing classes every 3 days. On which day will he have both classes together?

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Sam will have both the classes together on every 15th day.

Explanation

To find when he has both the class, we find the LCM of 5 and 3.


As both 5 and 3 are prime numbers, LCM (5, 3) is 5 × 3 = 15.

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Problem 3

If the LCM and product of two numbers are 175 and 875. Find the GCF of the numbers.

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The GCF of two numbers is 5.

Explanation

The GCF is calculated using the equation:


LCM × GCF = Product of the two numbers


Given, LCM = 175


Product of the numbers = 875


GCF = Product of two numbers / LCM


= 875 / 175

= 5
 

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Problem 4

If the LCM and product of two numbers are 84 and 336, find the GCF of the numbers.

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The GCF of the numbers is 4.

Explanation

The GCF is calculated using the equation:

LCM × GCF = Product of the two numbers

Given,

LCM = 84

Product of the numbers = 336

\( \mathrm{GCF} = \frac{\text{Product of two numbers}}{\mathrm{LCM}} \)

         = \( \frac{336}{84} \)

         = 4
 

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Problem 5

If the LCM and product of two numbers are 120 and 1080, find the GCF of the numbers.

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The GCF is 9.

Explanation

We use the formula,

\( \mathrm{GCF} = \frac{\text{Product of two numbers}}{\mathrm{LCM}} \)

Given,

Product of two numbers = 1080

LCM = 120

\( \mathrm{GCF} = \frac{1080}{120} = 9 \) 9

 

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FAQs on LCM

1.What is the LCM of 24 and 36?

The LCM of 24 and 36 is 72, as 72 is the smallest common multiple of 24 and 36.

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2.What is the LCM of 12 and 6?

The multiples of 12 are 12, 24, 36, 48,… And multiples of 6 are 6, 12, 18, 24,…

 

So, the LCM of 12 and 6 is 12.

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3.List the methods to calculate LCM.

The different methods to calculate the LCM are listing multiples, prime factorization, and division method. 
 

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4.What is the LCM of 5 and 11?

The LCM of 5 and 11 is 55. Since both 5 and 11 are prime numbers, the LCM is the product of 5 and 11.

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5.How to make a three-digit LCM?

Pick numbers whose LCM will be between 100 and 999. Then, find the LCM by multiplying the highest powers of all prime factors from those numbers.

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6.How to calculate the LCM of decimal numbers?

First, convert the decimals into whole numbers by removing the decimals. Then find the LCM of those whole numbers, and then adjust the result by placing the decimal point in the correct position.

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7.How to calculate the LCM of 3 numbers by division method?

First, write the numbers side by side, then divide each number by its standard prime factors. Bring down numbers that are not divisible, and continue the process until only one remains. Finally, multiply all the divisors to get the LCM.

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8.What is the LCM of 3 and 4?

The LCM of 3 and 4 is 12, because 12 is the smallest number.

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9.How to find the common denominator of 4 fractions?

It is found by determining the least common multiple (LCM) of their respective denominators.

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10.How can children in United States use numbers in everyday life to understand LCM (Least Common Multiple)?

Numbers appear everywhere—from counting money to measuring ingredients. Kids in United States see how LCM (Least Common Multiple) helps solve real problems, making numbers meaningful beyond the classroom.

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11.What are some fun ways kids in United States can practice LCM (Least Common Multiple) with numbers?

Games like board games, sports scoring, or even cooking help children in United States use numbers naturally. These activities make practicing LCM (Least Common Multiple) enjoyable and connected to their world.

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12.What role do numbers and LCM (Least Common Multiple) play in helping children in United States develop problem-solving skills?

Working with numbers through LCM (Least Common Multiple) sharpens reasoning and critical thinking, preparing kids in United States for challenges inside and outside the classroom.

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13.How can families in United States create number-rich environments to improve LCM (Least Common Multiple) skills?

Families can include counting chores, measuring recipes, or budgeting allowances, helping children connect numbers and LCM (Least Common Multiple) with everyday activities.

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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Fun Fact

: She loves to read number jokes and games.

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