By Neeraj K. Anand

Least Common Multiple LCM-Sainik School Class 6 Math Study Material Notes pdf download free-Sainik School Coaching Center In Jalandhar-Anand Classes

Least Common Multiple LCM-Sainik School Class 6 Math Study Material Notes pdf download free-Sainik School Coaching Center In Jalandhar-Anand Classes

Sainik School Entrance Exam for Class 6 Math Study Material Notes helps students face the competition in the current education system. In this case, the ANAND CLASSES is the best study tool to get a clear idea about the basics and gain a strong knowledge of the Sainik School Entrance Exam syllabus.

Least Common Multiple (LCM)

LCM stands for “Least Common Multiple.” It is a mathematical concept used to find the smallest multiple that two or more numbers share. To understand LCM from the basics, you should be familiar with a few key concepts:

Multiples: A multiple of a number is the result of multiplying that number by another integer. For example, the multiples of 3 are 3, 6, 9, 12, and so on.
Common Multiple: When you have two or more numbers, a common multiple is a number that is a multiple of each of those numbers. For example, consider the numbers 4 and 6. The common multiples of 4 and 6 are 12, 24, 36, and so on because each of these numbers is a multiple of both 4 and 6.

Now, let’s go through the process of finding the LCM of two or more numbers:

Step 1: Prime Factorization

Start by finding the prime factorization of each of the numbers. Prime factorization is the process of breaking down a number into its prime factors (prime numbers that multiply together to give the original number).
For example, let’s find the prime factorization of 12 and 18.
- The prime factorization of 12 is 2 * 2 * 3, which can be written as 2^2 * 3.
- The prime factorization of 18 is 2 * 3 * 3, which can be written as 2 * 3^2.

Step 2: Identify Common and Uncommon Prime Factors

Compare the prime factorizations of the numbers and identify the common prime factors and any unique prime factors for each number.
In our example, the common prime factor is 2, and the uncommon prime factors are 3 and 3^2 for 12 and 3 for 18.

Step 3: Multiply the Common and Uncommon Prime Factors

To find the LCM, you need to multiply the common prime factors and all the uncommon prime factors from all the numbers.
In our example, the LCM of 12 and 18 is calculated as follows: LCM(12, 18) = 2^2 * 3^2 * 3 = 4 * 9 * 3 = 36 * 3 = 108

So, the LCM of 12 and 18 is 108. This means 108 is the smallest multiple that both 12 and 18 have in common.

In summary, the LCM is the smallest multiple that two or more numbers share. To find it, you break down the numbers into their prime factors and then multiply together the common and uncommon prime factors.

Definition

Least Common Multiple(LCM) is a method to find the smallest common multiple between any two or more numbers. A common multiple is a number which is a multiple of two or more numbers.

Different Methods to find LCM

There are three important methods by which we can find the LCM of two or more numbers. They are:

Listing the Multiples
Prime Factorisation Method
Division Method

Let us learn here all three methods:

Listing the Multiples

The method to find the least common multiple of any given numbers is first to list down the multiples of specific numbers and then find the first common multiple between them.

Suppose there are two numbers 11 and 33. Then by listing the multiples of 11 and 33, we get;

Multiples of 11 = 11, 22, 33, 44, 55, ….

Multiples of 33 = 33, 66, 99, ….

We can see, the first common multiple or the least common multiple of both the numbers is 33. Hence, the LCM (11, 33) = 33.

LCM By Prime Factorisation

Another method to find the LCM of the given numbers is prime factorization. Suppose, there are three numbers 12, 16 and 24. Let us write the prime factors of all three numbers individually.

12 = 2 x 2 x 3

16 = 2 x 2 x 2 x 2

24 = 2 x 2 x 2 x 3

Now writing the prime factors of all the three numbers together, we get;

12 x 16 x 24 = 2 x 2 x 3 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3

Now pairing the common prime factors we get the LCM. Hence, there are four 2’s and one 3. So the LCM of 12, 16 and 24 will be;

LCM (12, 16, 24) = 2 x 2 x 2 x 2 x 3 = 48

LCM By Division Method

Finding LCM of two numbers by division method is an easy method. Below are the steps to find the LCM by division method:

First, write the numbers, separated by commas
Now divide the numbers, by the smallest prime number.
If any number is not divisible, then write down that number and proceed further
Keep on dividing the row of numbers by prime numbers, unless we get the results as 1 in the complete row
Now LCM of the numbers will be equal to the product of all the prime numbers we obtained in the division method

Let us understand with the help of examples.

Example: Find LCM of 10, 18 and 20 by division method.

Solution: Let us draw a table to divide the numbers by prime factors.

Prime factors	1st number	2nd number	3rd number
2	10	18	20
2	5	9	10
3	5	9	5
3	5	3	5
5	5	1	5
	1	1	1

Therefore, LCM (10, 18, 20) = 2 x 2 x 3 x 3 x 5 = 180

Now, if we have to find the common multiple of two or more numbers, then we have to write all the multiples for the given numbers. Say for example, if there are two numbers 4 and 6, then how to find the common multiple between them?

LCM of Two Numbers

Let us write multiples of 4 and 6 first,

4 : 4,8,12,16,20,24,28,…..

6: 6,12,18,24,30,36,42…..

From the above two expressions you can see, 4 and 6 have common multiples as 12 and 24. They may have more common multiple if we go beyond. Now, the smallest or least common multiple for 4 and 6 is 12. Therefore, 12 is the LCM of 4 and 6.

LCM Table

Numbers	LCM
24 and 36	72
10 and 15	30
8 and 10	40
15 and 20	60

LCM of Three Numbers

Now, let us take an example of 3 numbers.

Example: Find the LCM 4,6 and 12.

Solution: First write the common multiples of all three numbers.

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

Common Multiples of 6: 6,12,18,24,30,36,42…..

Common Multiples of 12: 12,24,36,48,60,72,….

From the above-given multiples of 4, 6 and 12, you can see, 12 is the smallest common multiple.

Therefore, LCM. of 4, 6 and 12 is 12.

Basics of LCM : Video Lecture L-1

Example 1: Find the LCM of 8 and 12.

Step 1: Prime Factorization

The prime factorization of 8 is 2 * 2 * 2, which can be written as 2^3.
The prime factorization of 12 is 2 * 2 * 3, which can be written as 2^2 * 3.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 2, and the uncommon prime factors are 2 and 3.

Step 3: Multiply the Common and Uncommon Prime Factors

To find the LCM, multiply the common prime factor and all the uncommon prime factors.
LCM(8, 12) = 2^3 * 2 * 3 = 8 * 2 * 3 = 48

So, the LCM of 8 and 12 is 48.

Example 2: Find the LCM of 15 and 20.

Step 1: Prime Factorization

The prime factorization of 15 is 3 * 5.
The prime factorization of 20 is 2 * 2 * 5, which can be written as 2^2 * 5.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 5, and the uncommon prime factors are 3 and 2^2.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(15, 20) = 5 * 3 * 2^2 = 5 * 3 * 4 = 60

So, the LCM of 15 and 20 is 60.

Example 3: Find the LCM of 24, 36, and 48.

Step 1: Prime Factorization

The prime factorization of 24 is 2 * 2 * 2 * 3, which can be written as 2^3 * 3.
The prime factorization of 36 is 2 * 2 * 3 * 3, which can be written as 2^2 * 3^2.
The prime factorization of 48 is 2 * 2 * 2 * 2 * 3, which can be written as 2^4 * 3.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 2, and the uncommon prime factors are 2, 2, 3, and 3^2.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(24, 36, 48) = 2^4 * 3^2 = 16 * 9 = 144

So, the LCM of 24, 36, and 48 is 144.

These examples illustrate how to find the LCM step by step by breaking down the numbers into their prime factors and then multiplying the common and uncommon prime factors to obtain the least common multiple.

Example 4: Find the LCM of 5, 6, and 10.

Step 1: Prime Factorization

The prime factorization of 5 is 5.
The prime factorization of 6 is 2 * 3.
The prime factorization of 10 is 2 * 5.

Step 2: Identify Common and Uncommon Prime Factors

There are no common prime factors among these numbers.
The uncommon prime factors are 2, 3, and 5.

Step 3: Multiply the Common and Uncommon Prime Factors

Since there are no common prime factors, the LCM is simply the product of all the prime factors present in these numbers.
LCM(5, 6, 10) = 2 * 3 * 5 = 30

So, the LCM of 5, 6, and 10 is 30.

Example 5: Find the LCM of 7, 14, and 21.

Step 1: Prime Factorization

The prime factorization of 7 is 7.
The prime factorization of 14 is 2 * 7.
The prime factorization of 21 is 3 * 7.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 7.
There are no uncommon prime factors.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(7, 14, 21) = 7 = 7

So, the LCM of 7, 14, and 21 is 7. In this case, they share a common prime factor, and that prime factor is the LCM itself.

Example 6: Find the LCM of 9, 12, and 18.

Step 1: Prime Factorization

The prime factorization of 9 is 3 * 3, which can be written as 3^2.
The prime factorization of 12 is 2 * 2 * 3, which can be written as 2^2 * 3.
The prime factorization of 18 is 2 * 3 * 3, which can be written as 2 * 3^2.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 3.
The uncommon prime factors are 2^2 and 2.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(9, 12, 18) = 3^2 * 2^2 * 2 = 9 * 4 * 2 = 72

So, the LCM of 9, 12, and 18 is 72.

Example 7: Find the LCM of 15, 25, and 35.

Step 1: Prime Factorization

The prime factorization of 15 is 3 * 5.
The prime factorization of 25 is 5 * 5, which can be written as 5^2.
The prime factorization of 35 is 5 * 7.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 5.
The uncommon prime factors are 3 and 7.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(15, 25, 35) = 5^2 * 3 * 7 = 25 * 3 * 7 = 525

So, the LCM of 15, 25, and 35 is 525.

Example 8: Find the LCM of 16, 24, and 32.

Step 1: Prime Factorization

The prime factorization of 16 is 2 * 2 * 2 * 2, which can be written as 2^4.
The prime factorization of 24 is 2 * 2 * 2 * 3, which can be written as 2^3 * 3.
The prime factorization of 32 is 2 * 2 * 2 * 2 * 2, which can be written as 2^5.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 2.
The uncommon prime factors are 3 and 2^3.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(16, 24, 32) = 2^5 * 3 * 2^3 = 32 * 3 * 8 = 256 * 3 = 768

So, the LCM of 16, 24, and 32 is 768.

Example 9: Find the LCM of 9, 10, and 15.

Step 1: Prime Factorization

The prime factorization of 9 is 3 * 3, which can be written as 3^2.
The prime factorization of 10 is 2 * 5.
The prime factorization of 15 is 3 * 5.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factors are 3 and 5.
There are no uncommon prime factors.

Step 3: Multiply the Common Prime Factors

LCM(9, 10, 15) = 3^2 * 5 = 9 * 5 = 45

So, the LCM of 9, 10, and 15 is 45.

Example 10: Find the LCM of 14, 28, and 42.

Step 1: Prime Factorization

The prime factorization of 14 is 2 * 7.
The prime factorization of 28 is 2 * 2 * 7, which can be written as 2^2 * 7.
The prime factorization of 42 is 2 * 3 * 7.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 7.
The uncommon prime factors are 2, 2, and 3.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(14, 28, 42) = 7 * 2^2 * 3 = 7 * 4 * 3 = 84

So, the LCM of 14, 28, and 42 is 84.

Example 11: Find the LCM of 20, 30, and 40.

Step 1: Prime Factorization

The prime factorization of 20 is 2 * 2 * 5, which can be written as 2^2 * 5.
The prime factorization of 30 is 2 * 3 * 5.
The prime factorization of 40 is 2 * 2 * 2 * 5, which can be written as 2^3 * 5.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 2.
The uncommon prime factors are 2, 2, 3, and 5.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(20, 30, 40) = 2^3 * 3 * 5 = 8 * 3 * 5 = 120

So, the LCM of 20, 30, and 40 is 120.

Example 12: Find the LCM of 27 and 36.

Step 1: Prime Factorization

The prime factorization of 27 is 3 * 3 * 3, which can be written as 3^3.
The prime factorization of 36 is 2 * 2 * 3 * 3, which can be written as 2^2 * 3^2.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 3.
The uncommon prime factors are 2^2.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(27, 36) = 3^3 * 2^2 = 27 * 4 = 108

So, the LCM of 27 and 36 is 108.

Example 13: Find the LCM of 16, 25, and 40.

Step 1: Prime Factorization

The prime factorization of 16 is 2 * 2 * 2 * 2, which can be written as 2^4.
The prime factorization of 25 is 5 * 5, which can be written as 5^2.
The prime factorization of 40 is 2 * 2 * 2 * 5, which can be written as 2^3 * 5.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 2.
The uncommon prime factors are 2 and 5^2.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(16, 25, 40) = 2^4 * 5^2 = 16 * 25 = 400

So, the LCM of 16, 25, and 40 is 400.

Example 14: Find the LCM of 5, 8, and 12.

Step 1: Prime Factorization

The prime factorization of 5 is 5.
The prime factorization of 8 is 2 * 2 * 2, which can be written as 2^3.
The prime factorization of 12 is 2 * 2 * 3, which can be written as 2^2 * 3.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 2.
The uncommon prime factors are 5 and 3.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(5, 8, 12) = 2^3 * 5 * 3 = 8 * 5 * 3 = 120

So, the LCM of 5, 8, and 12 is 120.

Example 15: Find the LCM of 14, 21, and 28.

Step 1: Prime Factorization

The prime factorization of 14 is 2 * 7.
The prime factorization of 21 is 3 * 7.
The prime factorization of 28 is 2 * 2 * 7, which can be written as 2^2 * 7.

Step 2: Identify Common and Uncommon Prime Factors

The common prime factor is 7.
The uncommon prime factors are 2 and 3.

Step 3: Multiply the Common and Uncommon Prime Factors

LCM(14, 21, 28) = 7 * 2^2 * 3 = 7 * 4 * 3 = 84

So, the LCM of 14, 21, and 28 is 84.

Frequently Asked Questions (FAQs) related to studying for the Sainik School Class 6 Math entrance exam

To help you prepare for Sainik School Class 6 Math, it’s important to use appropriate study materials. Here are some frequently asked questions (FAQ) related to studying for the Sainik School Class 6 Math entrance exam:

1. What is the syllabus for the Sainik School Class 6 Math entrance exam?

The syllabus for the entrance exam may vary slightly from one Sainik School to another. However, it generally covers topics from the standard Class 6 mathematics curriculum, including arithmetic, geometry, algebra, and basic mathematical concepts.

2. Where can I find official information about the exam pattern and syllabus?

You can find official information about the exam pattern and syllabus on the official website of the specific Sainik School you’re applying to. Each school may have its own admission criteria.

3. Which textbooks should I use for Class 6 Math preparation?

You should primarily use the NCERT Class 6 Math textbook. It covers the fundamental concepts and is widely accepted in Indian schools. Additionally, consider supplementary Math textbooks that are designed for competitive exams.

4. Are there any online resources for Sainik School Class 6 Math preparation?

Yes, you can find online resources such as video tutorials, practice questions, and mock tests on educational websites. Websites like ANAND CLASSES offer free Math materials that can be helpful for your preparation.

5. Where can I get sample papers and previous year’s question papers?

You can find sample papers and previous year’s question papers at online bookstore of ANAND CLASSES that sell competitive exam preparation materials. Additionally, ANAND CLASSES website offer downloadable PDFs of these papers for free or at a minimal cost.

6. Should I consider enrolling in coaching classes for Sainik School Math preparation?

Enrolling in coaching classes is a personal choice. While they can provide structured guidance and additional practice, they are not mandatory. You can achieve success through self-study and the use of appropriate study materials.

7. How should I manage my study time effectively for Class 6 Math preparation?

Create a study schedule that allocates specific time for Math preparation daily. Focus on understanding concepts, practicing regularly, and taking regular breaks to avoid burnout. Consistency is key.

8. Is there any specific advice for tackling the Math section of the Sainik School entrance exam?

Pay close attention to the BODMAS rule (Order of Operations) and practice solving a variety of math problems. Make sure you’re familiar with the types of questions that are commonly asked in the entrance exam.

Remember to check the specific requirements and guidelines provided by the Sainik School you are applying to, as these may vary from school to school.

To prepare effectively for the Sainik School Class 6 Math entrance exam, you should consider the following general sources:

Sainik School Official Website: Visit the official website of the Sainik School you are applying to. They often provide information about the exam pattern, syllabus, and sample question papers.
NCERT Textbooks: The National Council of Educational Research and Training (NCERT) textbooks are widely used in Indian schools and are a valuable resource for exam preparation. Ensure you have the NCERT Math textbook for Class 6.
Solved Sample Papers and Previous Year Question Papers: You can find solved sample papers and previous year question papers for Sainik School entrance exams at bookstores or online at ANAND CLASSES website. These papers can give you an idea of the exam pattern and types of questions asked.
Math Study Guides: The Math study guides and reference books are publish under publication department of ANAND CLASSES and are designed to help students prepare for entrance exams. Look for books specifically tailored to the Sainik School entrance exam.
Online Resources: www.anandclasses.co.in

We at ANAND CLASSES are providing notes for Sainik School Entrance Exam students, mainly for subjects like Science and Maths. Scoring well in these major subjects will increase the possibility of getting into good SAINIK SCHOOL in the long run. The notes that we are offering have been thoughtfully prepared by our experts to help you achieve the same. These notes are designed to help students overcome all the challenges in solving math problems and understand difficult MATH concepts. Basically, these notes act as a valuable reference tool for conducting effective revisions of the entire chapters given in each subject. Additionally, students can use these notes to get detailed explanations, practice problems, and study properly without wasting much precious time.

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