  # LCM Questions, Check Preparation Tip, Questions and Books

By BYJU'S Exam Prep

Updated on: September 25th, 2023 LCM, or Least Common Multiple, is a fundamental concept in mathematics that frequently appears in various competitive exams including SSC exams, aptitude tests, and problem-solving challenges. Understanding LCM and being able to solve related questions is essential for anyone aspiring to excel in mathematics specially in the Tier 1 and Tier 2 of CGL and CHSL exams.

In this article, we will explore different types of LCM questions, provide solutions, share tips to solve them efficiently, recommend helpful books, and offer preparation tips to enhance your LCM-solving abilities.

## LCM Questions: An Overview

In this section, we will delve into the basics of LCM questions, discussing what they are and why they are important. We will explore the concept of LCM, its definition, and its relevance in real-life scenarios. Additionally, we will highlight the significance of LCM in various mathematical operations and problem-solving techniques that can be used in CGL exam too.

The Least Common Multiple (LCM) of two or more integers refers to the smallest multiple that is divisible by all the integers involved. For instance, 20 is a multiple of 5 because 5 × 4 = 20. Therefore, 20 is divisible by both 5 and 4. Since 10 is the smallest positive integer divisible by both 5 and 4, it is the LCM of 5 and 4.

To understand LCM better, let’s consider an example:

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70…

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72…

Common multiples of 5 and 6: 30, 60, 90, 120…

Hence, the lowest common multiple is the first number in the list of common multiples, which is 30.

## Method to Calculate L.C.M

There are different methods of calculating LCM of the numbers. Let us have a look at them one by one

By Prime Factorizations

According to the prime factorization theorem, every positive integer greater than 1 can be expressed uniquely as a product of prime numbers.

Let’s find the LCM of 9, 48, and 21:

1. Find the factors of each number and express them as a product of prime numbers’ powers.
– 9 =

3^2
– 48 = 2^4 * 3
– 21 = 3 * 7

2. Write all the factors with their highest powers: 2^4, 3^2, and 7.

3. Multiply these factors to obtain the LCM: 2^4 * 3^2 * 7 = 1008.

By Division Method

In this method, divide all the integers by a common number until no two numbers are further divisible. Then, multiply the common divisor by the remaining number to obtain the LCM.

## H.C.F.: An Overview

Multiplication is a familiar operation to us all, such as 2 * 3 = 6. However, the Highest Common Factor (HCF) involves the reverse process, known as factorization. Factorization breaks down a composite number into its prime factors. For example, 6 = 2 * 3, where 6 is a composite number and 2 and 3 are prime numbers.

In mathematics, the Highest Common Factor (HCF) refers to the largest positive integer that divides two or more numbers without leaving a remainder. To illustrate, the HCF of 8 and 12 is 4.

## Method to Calculate H.C.F

There are different methods of calculating HCF of the numbers. Let us have a look at them one by one

By Prime Factorizations

To calculate the Highest Common Factor, we first determine the prime factors of the two numbers and then identify the common factors.

For instance, let’s consider the numbers 18 and 42. The prime factors of 18 are 2 * 3 * 3, and the prime factors of 42 are 7 * 2 * 3. By observing the common factors, we find that they are 2 and 3. Therefore, the HCF of 18 and 42 is 6.

By Division Method

In this method, we start by dividing the higher number by the smaller number.

1. Place the higher number as the dividend and the smaller number as the divisor.
2. Divide and obtain the remainder. Use this remainder as the new divisor and the earlier divisor as the dividend.
3. Repeat this process until you get a remainder of zero. The last divisor obtained is the HCF.
4. If there are more than two numbers, continue the process by dividing the third lowest number by the last divisor obtained in the previous steps.

## HCF and LCM Questions

HCF (Highest Common Factor) and LCM are closely related concepts. Understanding their interplay is crucial for tackling more advanced mathematical problems in the Staff Selection Commission (SSC) exams. In this section, we will explore the connection between HCF and LCM, discuss common questions that involve both concepts, and provide strategies to solve them effectively. By mastering the relationship between HCF and LCM, you will be better equipped to handle complex problem-solving scenarios.

### The Relation between L.C.M. and H.C.F. of Two Natural Numbers

The product of the LCM and HCF of two natural numbers is equal to the product of the numbers themselves.

For example:

LCM (8, 28) = 56 and HCF (8, 28) = 4

Now, 8 * 28 = 224, and likewise, 56 * 4 = 224.

Formulae to Find H.C.F. and L.C.M. of Fractions

To find the HCF and LCM of a fractional number, we use the following formulas:

• HCF of fraction = HCF of numerator / LCM of denominator
• LCM of fraction = LCM of numerator / HCF of denominator.

## LCM Questions with Solutions

Here, we will provide a comprehensive set of LCM questions with step-by-step solutions. These questions will cover a wide range of difficulty levels and scenarios, enabling you to practice and reinforce your understanding of LCM. By examining the solutions, you will gain valuable insights into the techniques and strategies involved in tackling different types of LCM problems.

1. Find the greatest number less than 900 that is divisible by 8, 12, and 28.

A) 845
B) 840
C) 850
D) 835

Solution: The least number divisible by 8, 12, and 28 is 168. Any multiple of 168 will be divisible by each of these numbers. Since we need to find the greatest number below 900, the largest multiple of 168 below 900 is 168 * 5 = 840.

2. The sum of two numbers is PQ, and their difference is 1/7th of their sum. Find their HCF.

A)
B)
C)
D)

Solution: Let the two numbers be a and b.

Given:
a + b = PQ …………(i)
a – b = (1/7)(a + b) ………….(ii)

From equations (i) and (ii):
2b = (6/7)(a + b)
7b = 6a + 6b
b = (6/7)a

Substituting the value of b in equation (i):
a + (6/7)a = PQ
(13/7)a = PQ
a = (7/13)PQ

The HCF of a and b will be the HCF of (7/13)PQ and (6/7)PQ, which is PQ itself.

3. Five cement pipes of lengths 210 cm, 225 cm, 285 cm, 315 cm, and 345 cm are to be cut into equal-length pieces. Calculate the maximum number of pieces that can be cut.

A) 80
B) 85
C) 92
D) 95

Solution: The maximum number of pieces that can be cut will be equal to the highest common factor (HCF) of the given lengths.

HCF of (210, 225, 285, 315, 345) = 15

Number of parts from 210 = 210/15 = 14
Number of parts from 225 = 225/15 = 15
Number of parts from 285 = 285/15 = 19
Number of parts from 315 = 315/15 = 21
Number of parts from 345 = 345/15 = 23

Maximum number of pieces that can be cut = 14 + 15 + 19 + 21 + 23 = 92

4. Find the largest two-digit number that divides 211 and 396, giving the same remainder.

A) 37
B) 185
C) 43
D) 21

Solution: The difference between the numbers is 185. The required number will be the factor of the difference. Thus, the factor of 185 is 37, which divides both 211 and 396, giving the same remainder.

5. Four bells ring at intervals of 8, 12, 16, and 28 seconds, starting simultaneously at 1:00 ‘o’ clock. At what time will they ring simultaneously again?

A) 2:03:23 o’clock
B) 2:02:36 o’clock
C) 1:05:36 o’clock
D) 1:05:48 o’clock

Solution: The prime factorization of the given intervals is:
8 = 2 × 2 × 2
12 = 2 × 2 × 3
16 = 2 × 2 × 2 × 2
28

= 2 × 2 × 7

The least common multiple (LCM) of 8, 12, 16, and 28 is 2 × 2 × 2 × 3 × 2 × 7 = 336 seconds, which is equivalent to 5 minutes 36 seconds.

Since the bells start ringing at 1:00 o’clock, they will ring together again at 1:05:36 o’clock.

In addition to the above given questions, candidates must practice number system questions for SSC exams to score better.

## Preparation Tips for LCM Questions

To excel in LCM questions, it is important to adopt a structured approach to your preparation for the SSC MTS 2023. In this final section, we will provide valuable tips and guidelines to help you prepare effectively for LCM-related challenges. From building a strong foundation to practicing regularly and managing your time efficiently, these tips will set you on the path to success in LCM problem-solving.

• Learn the fundamentals of prime factorization, LCM, and the divisibility rule.
• Practice answering a range of LCM questions with varying degrees of difficulty. You can check your grasp over the subject by attempting the sectional quiz on LCM and HCF.
• To learn the format and kinds of LCM problems asked in examinations, solve previous years’ SSC exam papers.
• To increase your speed and accuracy during practice Quant Quizzes, time yourself.
• By attempting LCM-related questions and SSC exams mock tests, you may regularly improve and solidify your conceptual knowledge.

## Tips to Solve LCM Questions Efficiently

Solving LCM questions can sometimes be challenging, especially when dealing with complex numbers or multiple variables. In this section, we will share practical tips and techniques to approach LCM questions more efficiently. From prime factorization to the “cake method” and other shortcuts, these strategies will empower you to solve LCM problems accurately and in less time.

• Prime Factorization Method: Break down each number into its prime factors. Then, take the highest power of each prime factor that appears in any of the numbers, and multiply them together. The product will be the LCM. This method is particularly useful when dealing with large numbers or when finding the LCM of more than two numbers.
• List Method: Create a list of multiples for each number until you find a common multiple. Start with the given numbers and keep adding their multiples until you find the smallest number that appears in all the lists. That number will be the LCM. This method is helpful when dealing with small numbers or when finding the LCM of only two numbers.
• Elimination Method: Find the LCM of two numbers and then use it as one of the numbers to find the LCM with the next number. Repeat this process until you have considered all the given numbers. This method is suitable when you have multiple numbers and want to find their LCM step by step.
• Prime Factorization Table: Create a table that includes the prime factorization of each number. Then, identify the highest power of each prime factor that appears in any of the numbers. Multiply these powers together to obtain the LCM. This method helps organize the prime factorization process and can be especially useful for solving complex LCM problems.
• Simplify the Numbers: If the given numbers have common factors, simplify them by dividing each number by their highest common factor (HCF). Then, find the LCM of the simplified numbers. This approach reduces the complexity of the problem by working with smaller numbers.

Remember, practice is essential for becoming proficient in solving LCM questions efficiently. The more you practice using these techniques, the faster and more accurate you’ll become.

## Recommended Books for LCM Questions

To further enhance your LCM-solving skills, we recommend following books that focus specifically on LCM questions.

• “Fast Track Objective Arithmetic” by Rajesh Verma
• “Quantitative Aptitude for Competitive Examinations” by R.S. Aggarwal
• “Magical Book on Quicker Maths” by M. Tyra

These resources provide comprehensive coverage of LCM concepts, examples, and practice exercises. By referring to these books, you can deepen your understanding of LCM and gain exposure to a variety of LCM questions, ultimately boosting your confidence and proficiency.

Conclusion

LCM questions are an integral part of various mathematics examinations, and having a strong grasp of this concept is essential for success. By studying the basics, practicing with a variety of LCM questions, and applying effective problem-solving strategies, you can sharpen your skills and approach LCM questions with confidence. Remember to refer to recommended books and incorporate preparation tips to maximize your potential. You can also join our Online classroom program to crack SSC Exams with flying colors. With dedication and consistent effort, you’ll be well on your way to mastering LCM questions and achieving your mathematical goals.

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