LCM (Least Common Multiple)

Introduction to LCM

LCM stands for Least Common Multiple. It is one of the most important concepts in mathematics. It defines the smallest multiple of two or more numbers that are in common. For example, if you want to find the LCM of 3 and 9, you have to look for the smallest number that appears in the lists of multiples of both 3 and 6, which in this case is 3. There are different methods to find the LCM, which include listing out the multiples, using prime factorisation, or using the long division method. Learning LCM is necessary because it helps solve problems involving fractions, ratios, and time intervals. In this article we will explain the different methods of how to calculate LCM along with LCM formulas and examples to help you master the concept of LCM easily.

 

Table of Contents

• What is LCM?
• Why is LCM Important?
• Methods of Calculating LCM
• Listing Multiples Method
• Prime Factorization Method
• Division Method (Common Division)
• LCM Formula Using HCF
• How to Find LCM of Three Numbers
• How to Calculate LCM of Fractions
• Common Mistakes When Finding LCM
• LCM Solved Examples
• Conclusion
• FAQs on LCM

 

What is LCM?

The Least Common Multiple, or LCM, is the smallest positive number that is a multiple of two or more specified numbers. In other words, it is the smallest number that does not result in a remainder when divided by all other numbers. For example, the LCM of 4 and 5 is 20 because it is the smallest number that is exactly divisible by both 4 and 5. The LCM can be found by listing the multiples of each number and locating the first common multiple, or by using methods such as prime factorization or the LCM formula involving the HCF (Highest Common Factor).

Definition of LCM:

The LCM is the lowest positive number that can be divided exactly by two or more given numbers.

Example:

LCM of 3 and 4 is 12 because 12 is the smallest number divisible by both 3 and 4.

 

Why is LCM Important?

Understanding how to calculate LCM helps in:

• Fractions with different denominators added and subtracted

• Resolving issues with time, schedules, and repetition

• Using algebraic equations, ratios, and proportions

• Effectively solving problems with tests and everyday life

Learning how to find LCM makes a variety of calculations easier.

 

Methods of Calculating LCM

The LCM (Least Common Multiple) of two or more numbers can be calculated in several efficient ways. Depending on the situation and the size of the numbers, each approach has its uses. The most common methods are listed below:

There are three primary methods for how to find LCM:

1. Listing Multiples Method

2. Prime Factorization Method

3. Division Method (Common Division)

Each method is useful based on the type of numbers involved. Let’s explore them with detailed LCM examples.

 

1. Listing Multiples Method

The Listing Multiples Method is the most basic and beginner-friendly technique to calculate the LCM (Least Common Multiple). It works best for small numbers and helps build a clear understanding of what a multiple is.

Steps to Follow:

1. Write down the first few multiples of each number.

2. Identify the common multiples in the lists.

3. Choose the smallest (least) of these common multiples,that is your LCM.

Example 1: Find the LCM of 4 and 5

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28...

• Multiples of 5: 5, 10, 15, 20, 25, 30...

Common multiple: 20

LCM(4, 5) = 20

Example 2: Find the LCM of 3 and 6

• Multiples of 3: 3, 6, 9, 12, 15...

• Multiples of 6: 6, 12, 18, 24...

Common multiple: 6

LCM(3, 6) = 6

 

2. Prime Factorization Method

The Prime Factorization Method is a systematic and accurate way to find the LCM (Least Common Multiple), especially for larger numbers. It involves expressing each number as a product of its prime factors, and then using those factors to calculate the LCM.

Steps to Follow:

1. Perform prime factorization of each number (break the number into its prime factors).

2. Identify all the prime numbers that appear in the factorizations.

3. For each prime number, take the highest power that appears in any of the factorizations.

4. Multiply all the highest powers of the primes to get the LCM.

Example 1: Find the LCM of 12 and 18

Step 1: Prime Factorization

• 12 = 2² × 3

• 18 = 2 × 3²

Step 2: Take the highest powers of each prime

• 2² (from 12)

• 3² (from 18)

Step 3: Multiply

LCM = 2² × 3² = 4 × 9 = 36

One of the most dependable methods for learning how to compute LCM is the Prime Factorization Method, particularly when working with large values or three or more numbers.

 

3. Division Method (Common Division)

The Division Method, also known as the Common Division Method, is an organized and efficient way to find the LCM (Least Common Multiple) of two or more numbers, especially when dealing with larger sets. This method involves dividing all the numbers simultaneously by their common prime factors until all results become 1.

Steps to Follow:

1. Write the numbers in a row.

2. Divide all the numbers by a common prime number (if divisible).

3. Write the quotient below each number.

4. Repeat the division process until all resulting numbers are 1.

5. Multiply all the prime divisors used -the result is the LCM.
   
   

Example 1: Find the LCM of 24 and 36

Let’s divide both numbers step by step:

Now multiply all the divisors:

2 × 2 × 3 × 2 × 3 = 72

So, LCM(24, 36) = 72

 

Example 2: Find the LCM of 15, 20, and 30

Now multiply the prime divisors:

2 × 2 × 3 × 5 = 60

So, LCM(15, 20, 30) = 60

When working with more than two numbers, the Division Method is a quick and easy way to understand how to calculate LCM. It avoids the need for separate factorizations and keeps all steps visible in a tabular form.

 

LCM Formula Using HCF

Finding the LCM (Least Common Multiple) of two numbers can be done quickly and easily by using their HCF(Highest Common Factor). When you already know how to calculate the HCF of two numbers, this method is very helpful.

Formula:

LCM(a, b) = (a × b) / HCF(a, b)

This formula shows that:

LCM × HCF = a × b

Steps to Use the Formula:

1. Multiply the two numbers together.

2. Find the HCF of the two numbers.

3. Divide the product by the HCF to get the LCM.

Example 1: LCM of 15 and 20

Step 1: 15 × 20 = 300

Step 2: HCF of 15 and 20 is 5

Step 3: LCM = 300 ÷ 5 = 60

So, LCM(15, 20) = 60

Example 2: LCM of 18 and 24

Step 1: 18 × 24 = 432

Step 2: HCF of 18 and 24 is 6

Step 3: LCM = 432 ÷ 6 = 72

So, LCM(18, 24) = 72

 

How to Find the LCM of Three Numbers

You don't need a new technique to determine the LCM (Least Common Multiple) of three numbers. Rather, you use any LCM method or the LCM formula, two numbers at a time, step-by-step. This prevents misunderstandings and streamlines the procedure.

Step-by-Step Process:

Step 1: Find the LCM of the first two numbers.

Step 2: Use the result and find the LCM with the third number.

Step 3: The final answer is the LCM of all three numbers.

Formula:

LCM(a, b, c) = LCM(LCM(a, b), c) 

Example: LCM of 3, 4, and 6

Step 1: LCM of 3 and 4 = 12

Step 2: LCM of 12 and 6 = 12

Final Answer: LCM(3, 4, 6) = 12

 

How to Calculate LCM of Fractions

A unique rule that combines the LCM of numerators and the HCF (Highest Common Factor) of denominators is required to determine the LCM (Least Common Multiple) of fractions.

LCM of Fractions Formula:

LCM of fractions = (LCM of numerators) / (HCF of denominators)

Steps to Calculate LCM of Fractions:

1. Find the LCM of the numerators of the fractions.

2. Find the HCF of the denominators of the fractions.

3. Divide the LCM of numerators by the HCF of denominators.

Example: LCM of 2/5 and 3/10

Step 1: LCM of numerators (2 and 3)

LCM(2, 3) = 6

Step 2: HCF of denominators (5 and 10)

HCF(5, 10) = 5

Step 3: Apply the formula

LCM = 6 / 5

So, LCM(2/5, 3/10) = 6/5

 

Common Mistakes When Finding LCM

• Confusing LCM with HCF

• Not using the highest powers of primes

• Missing common multiples

• Miscalculating in the listing method

Double-check your steps, especially when using prime factorization or formulas.

 

LCM Solved Examples

Example 1: LCM of 8 and 12

8 = 2³

12 = 2² × 3

LCM = 2³ × 3 = 24

Example 2: LCM of 9 and 6

9 = 3²

6 = 2 × 3

LCM = 2 × 3² = 18

Example 3: LCM of 10, 15, and 20

First, LCM(10, 15) = 30

Then, LCM(30, 20) = 60

Final LCM = 60

 

Conclusion

Understanding how to find the LCM or least common multiple is essential skill for students and professionals alike.  It is a foundational skill applied in solving a wide variety of mathematical and practical questions. You can solve more HCF and LCM questions to build conceptual clarity on this topic. Once you understand how to calculate LCM using the division method, prime factorization or listing multiples you will be able to solve many questions based on advanced topics related to it.

 
Frequently Asked Questions on LCM

1. What is the LCM for 24 and 36?

Ans: To find the LCM of 24 and 36, use prime factorization:

• 24 = 2³ × 3

• 36 = 2² × 3²

Take the highest powers of all primes:

LCM = 2³ × 3² = 8 × 9 = 72

So, LCM(24, 36) = 72

 

2. What's the LCM of 12 and 18?

Ans: Prime factorization method:

• 12 = 2² × 3

• 18 = 2 × 3²

LCM = 2² × 3² = 4 × 9 = 36

So, LCM(12, 18) = 36

 

3. How is LCM calculated?

Ans: LCM (Least Common Multiple) can be calculated using several methods:

• Listing multiples method

• Prime factorization method

• Common division (division method)

• LCM formula using HCF:
  LCM(a, b) = (a × b) / HCF(a, b)

Each method finds the smallest number that is divisible by all given numbers.

 

4. What is the LCM of 8 and 10?

Ans: Prime factorization:

• 8 = 2³

• 10 = 2 × 5

LCM = 2³ × 5 = 8 × 5 = 40

So, LCM(8, 10) = 40

 

5. What is the LCM calculator?

Ans: An LCM calculator is a digital tool (online or offline) used to quickly compute the Least Common Multiple of two or more numbers. You simply enter the numbers, and the calculator shows the result instantly. It's helpful for checking your answers or saving time on large numbers. Many websites and math apps offer free LCM calculators.

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