How to Find Least Common Multiple (LCM) Using Factorization | Real World Problems of Factorization

How to Find Least Common Multiple (LCM)?

Finding the Least Common Multiple (LCM) is an important skill in mathematics. LCM is used to solve many mathematical and real-life problems involving repeated events, common schedules, fractions, and numbers that need to occur together.

One of the easiest methods for finding LCM is prime factorization. Understanding the use of factorization, how to use factorization, and the application of factorization can make LCM problems much easier.

In this article, we will learn how to find LCM using prime factorization, understand the steps involved, and solve some real world problems of factorization.

What is a Multiple?

A multiple of a number is obtained by multiplying that number by a positive integer.

For example, the multiples of 4 are:

4, 8, 12, 16, 20, 24, 28, 32, …

The multiples of 6 are:

6, 12, 18, 24, 30, 36, …

The common multiples of 4 and 6 are:

12, 24, 36, …

The smallest positive common multiple is 12.

Therefore:

LCM of 4 and 6 = 12

$A=4$A=4$B=6$B=6048120612

$\text{LCM}=12$LCM=12

$\text{LCM}(4,6)=12$LCM(4,6)=12

3 jumps of 4 and 2 jumps of 6 meet at 12

$A$A

$A$A

$B$B

$B$B

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What is Least Common Multiple?

The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all the given numbers.

For example:

Multiples of 3:

3, 6, 9, 12, 15, 18, …

Multiples of 4:

4, 8, 12, 16, 20, …

The first common multiple is 12.

Therefore:

LCM(3, 4) = 12

Why is Factorization Useful for Finding LCM?

Factorization means expressing a number as a product of its factors.

For example:

12 = 2 × 2 × 3

or

12 = 2² × 3

Prime factorization provides a systematic way to find the LCM of numbers, especially when the numbers are large.

The main use of factorization in finding LCM is that it allows us to identify the prime factors of each number and select the greatest power of every prime factor.

How to Use Factorization to Find LCM?

Follow these simple steps:

Step 1: Find the prime factorization of each number

Suppose we want to find the LCM of 12 and 18.

Prime factorization of 12:

12 = 2² × 3

Prime factorization of 18:

18 = 2 × 3²

Step 2: Identify all prime factors

The prime factors are:

2 and 3

Step 3: Select the highest power of each prime factor

For 2:

Highest power = 2²

For 3:

Highest power = 3²

Step 4: Multiply the highest powers

LCM = 2² × 3²

LCM = 4 × 9

LCM = 36

Therefore:

LCM of 12 and 18 = 36

Another Example of Finding LCM by Factorization

Find the LCM of 20 and 30.

Prime factorization of 20:

20 = 2² × 5

Prime factorization of 30:

30 = 2 × 3 × 5

Take the highest power of every prime factor:

2², 3 and 5

Therefore:

LCM = 2² × 3 × 5

LCM = 4 × 3 × 5

LCM = 60

So:

LCM(20, 30) = 60

How to Find LCM of Three Numbers?

The same method can be used for three or more numbers.

Find the LCM of 8, 12 and 18.

Prime factorization:

8 = 2³

12 = 2² × 3

18 = 2 × 3²

Take the highest power of each prime:

2³ and 3²

Therefore:

LCM = 2³ × 3²

LCM = 8 × 9

LCM = 72

Hence:

LCM of 8, 12 and 18 = 72

Real World Problems of Factorization

Factorization is not only useful for solving textbook questions. There are several real world problems of factorization where factors and multiples help us find common timings, schedules, and repeated events.

Example 1: School Bells

Two school bells ring at different intervals. One bell rings every 6 minutes and another rings every 8 minutes.

After how many minutes will both bells ring together again?

We need to find:

LCM of 6 and 8.

6 = 2 × 3

8 = 2³

LCM = 2³ × 3

LCM = 8 × 3

LCM = 24

Therefore, both bells will ring together again after 24 minutes.

This is an example of the application of factorization in a real-life situation.

Example 2: Bus Schedules

Suppose one bus arrives at a stop every 12 minutes and another bus arrives every 18 minutes. If both buses arrive together now, after how many minutes will they arrive together again?

Find the LCM of 12 and 18.

12 = 2² × 3

18 = 2 × 3²

LCM = 2² × 3²

LCM = 36

Therefore, the buses will arrive together again after 36 minutes.

Application of Factorization in Mathematics

The application of factorization is much broader than simply finding LCM.

Factorization can be used for:

• Finding LCM
• Finding HCF/GCD
• Simplifying mathematical expressions
• Solving some algebraic equations
• Working with fractions
• Identifying prime factors
• Solving problems involving repeated events

Learning how to use factorization gives students a strong foundation for many topics in mathematics.

LCM and HCF: Important Difference

Students sometimes confuse LCM and HCF.

LCM means Least Common Multiple. It is the smallest positive number that is divisible by all the given numbers.

HCF means Highest Common Factor. It is the greatest factor common to all the given numbers.

For example, for 12 and 18:

HCF = 6

LCM = 36

So remember:

HCF → common factor

LCM → common multiple

Important Rules for Finding LCM by Prime Factorization

Remember these points:

1. First find the prime factorization of every number.
2. Write each number as a product of prime factors.
3. List every prime factor that occurs.
4. Choose the highest power of each prime factor.
5. Multiply those highest powers.
6. The result is the LCM.

Common Mistakes Students Should Avoid

Students often make these mistakes while finding LCM:

• Choosing the smallest power instead of the highest power.
• Forgetting a prime factor that appears in one of the numbers.
• Confusing LCM with HCF.
• Making errors in prime factorization.
• Multiplying all the prime factors from every number instead of selecting the highest power of each.

Short Notes on Finding LCM

Least Common Multiple (LCM): The smallest positive common multiple of two or more numbers.

Prime Factorization: Expressing a number as a product of prime numbers.

Method to Find LCM:

1. Find prime factorization of each number.
2. Write all different prime factors.
3. Select the highest power of each prime factor.
4. Multiply them together.

Example:

12 = 2² × 3

18 = 2 × 3²

Therefore:

LCM = 2² × 3² = 36

Key idea:

For LCM, take the highest power of every prime factor.

MCQs on Finding LCM

1. What does LCM stand for?

A. Lowest Common Factor
B. Least Common Multiple
C. Largest Common Multiple
D. Least Factor Multiple

Answer: B. Least Common Multiple

2. What is the LCM of 4 and 6?

A. 8
B. 10
C. 12
D. 24

Answer: C. 12

3. What is the prime factorization of 12?

A. 2 × 6
B. 3 × 4
C. 2² × 3
D. 2 × 3

Answer: C. 2² × 3

4. What is the LCM of 12 and 18?

A. 24
B. 30
C. 36
D. 54

Answer: C. 36

5. When finding LCM by prime factorization, we select:

A. Lowest power of each prime
B. Highest power of each prime
C. Only even factors
D. Only common factors

Answer: B. Highest power of each prime

6. What is the prime factorization of 20?

A. 2² × 5
B. 2 × 10
C. 4 × 5
D. 2 × 5

Answer: A. 2² × 5

7. What is the LCM of 8 and 12?

A. 12
B. 16
C. 24
D. 36

Answer: C. 24

8. Which mathematical concept is useful for finding LCM?

A. Prime factorization
B. Measurement only
C. Geometry only
D. Graphing only

Answer: A. Prime factorization

9. What is the LCM of 5 and 10?

A. 5
B. 10
C. 15
D. 50

Answer: B. 10

10. Which of the following is an application of LCM?

A. Finding when repeated events occur together
B. Measuring temperature only
C. Drawing circles only
D. Finding angles only

Answer: A. Finding when repeated events occur together

Worksheet / Assignment

Part A: Find the LCM

Find the LCM of the following numbers using prime factorization.

1. 6 and 9
2. 8 and 12
3. 10 and 15
4. 12 and 20
5. 15 and 25
6. 18 and 24
7. 20 and 30
8. 16 and 24
9. 18 and 30
10. 12, 18 and 24

Part B: Real-Life Problems

Question 1

A bell rings every 8 minutes and another bell rings every 12 minutes. If both bells ring together now, after how many minutes will they ring together again?

Question 2

One bus arrives at a stop every 10 minutes and another every 15 minutes. If they arrive together at 8:00 a.m., when will they next arrive together?

Question 3

A machine completes one cycle every 6 minutes and another machine completes a cycle every 9 minutes. After how many minutes will both machines complete a cycle at the same time?

Question 4

Three lights flash at intervals of 4, 6 and 8 seconds. If they flash together now, after how many seconds will they flash together again?

Question 5

Explain in your own words how prime factorization helps us find the LCM.

Challenge Question

Find the LCM of:

24, 36 and 54

Show all the steps using prime factorization.

Conclusion

The Least Common Multiple (LCM) is an important concept in mathematics and can be found easily using prime factorization. To find LCM, factorize each number into prime factors, select the highest power of every prime factor, and multiply them together.

Understanding the use of factorization, how to use factorization, and the application of factorization helps students solve both mathematical exercises and real world problems of factorization involving schedules, repeated events, and common timings.

Students who want to strengthen their Class 9 Mathematics preparation can also explore the recorded lectures, notes, assessments, and other learning resources available at Nisar Math Academy.

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