LCM And HCF Problems With Solutions

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#Introduction

In the realm of mathematics, understanding the relationships between numbers is paramount. Among the fundamental concepts that govern these relationships are the Least Common Multiple (LCM) and the Highest Common Factor (HCF). These concepts serve as the building blocks for more advanced mathematical principles and play a crucial role in solving a myriad of numerical problems. In this comprehensive guide, we delve into the intricacies of LCM and HCF, exploring their definitions, properties, and applications. We will also unravel the fascinating interplay between these concepts and number relationships, equipping you with the tools to tackle complex mathematical challenges with confidence.

Defining LCM and HCF: The Cornerstones of Number Relationships

At the heart of our exploration lie the definitions of LCM and HCF. The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all of the given numbers. In simpler terms, it is the smallest number that appears in the multiples of each of the given numbers. For instance, the LCM of 4 and 6 is 12, as 12 is the smallest number that is divisible by both 4 and 6.

Conversely, the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest positive integer that divides all of the given numbers without leaving a remainder. In essence, it is the largest number that is a factor of each of the given numbers. For example, the HCF of 12 and 18 is 6, as 6 is the largest number that divides both 12 and 18.

Unveiling the Interplay between LCM and HCF: A Symbiotic Relationship

LCM and HCF are not merely isolated concepts; they share a deep-rooted connection that forms the cornerstone of number relationships. This connection is elegantly captured by the following fundamental formula:

LCM (a, b) × HCF (a, b) = a × b

where 'a' and 'b' represent the two numbers under consideration. This formula reveals a profound relationship: the product of the LCM and HCF of two numbers is equal to the product of the numbers themselves. This principle serves as a powerful tool for solving problems involving LCM and HCF, allowing us to deduce one quantity if the others are known.

Let's embark on a problem-solving journey to solidify our understanding of LCM, HCF, and their interplay. Consider the following scenario:

The LCM of two numbers is 14 times their HCF. The sum of the LCM and HCF is 600. If one number is 280, then find the other number.

Cracking the Code: A Step-by-Step Solution

To unravel this numerical puzzle, we will employ a systematic approach, breaking down the problem into manageable steps.

  1. Introducing Variables: Let's denote the LCM as 'L' and the HCF as 'H'. According to the problem statement, we have L = 14H.

  2. Formulating Equations: We are also given that the sum of the LCM and HCF is 600, which translates to the equation L + H = 600.

  3. Substitution and Simplification: Substituting L = 14H into the equation L + H = 600, we get 14H + H = 600, which simplifies to 15H = 600.

  4. Solving for HCF: Dividing both sides of the equation 15H = 600 by 15, we find H = 40.

  5. Determining LCM: Substituting H = 40 into the equation L = 14H, we obtain L = 14 × 40 = 560.

  6. Applying the Fundamental Formula: Now, we invoke the fundamental formula LCM (a, b) × HCF (a, b) = a × b. We know that one number (a) is 280, and we have calculated the LCM (L = 560) and HCF (H = 40). Let the other number be 'b'.

  7. Solving for the Unknown Number: Plugging in the values, we get 560 × 40 = 280 × b. Solving for 'b', we find b = (560 × 40) / 280 = 80.

Therefore, the other number is 80.

The Answer Revealed: Option (c) 80

Thus, the correct answer to this problem is (c) 80. By systematically applying the concepts of LCM, HCF, and their interrelationship, we have successfully navigated the numerical puzzle and arrived at the solution.

Let's delve into another problem that showcases the versatility of HCF and its applications in expressing numerical relationships. Consider the following scenario:

The HCF of 408 and 1032 is expressible in the form 1032m - 2040. Find the value of m.

Unlocking the Solution: A Journey Through HCF Properties

To unravel this problem, we will leverage the properties of HCF and explore how it can be expressed in different forms.

  1. Calculating the HCF: The first step is to determine the HCF of 408 and 1032. We can employ the Euclidean algorithm, a systematic method for finding the HCF of two numbers. Applying the Euclidean algorithm, we find that the HCF of 408 and 1032 is 24.

  2. Equating the Expressions: According to the problem statement, the HCF of 408 and 1032 can be expressed in the form 1032m - 2040. Since we have calculated the HCF to be 24, we can equate the two expressions: 1032m - 2040 = 24.

  3. Solving for m: Now, we need to solve for the value of 'm'. Adding 2040 to both sides of the equation, we get 1032m = 2064.

  4. Isolating m: Dividing both sides of the equation by 1032, we find m = 2064 / 1032 = 2.

Therefore, the value of m is 2.

The Solution Unveiled: Option (a) 2

The correct answer to this problem is (a) 2. By understanding the properties of HCF and employing algebraic manipulation, we have successfully determined the value of 'm' that satisfies the given condition.

LCM and HCF are not merely abstract mathematical concepts confined to textbooks; they have practical applications in various real-world scenarios. Let's explore a few examples:

  1. Scheduling and Synchronization: LCM is instrumental in scheduling events that occur at regular intervals. For instance, if two buses depart from the same station at intervals of 15 minutes and 20 minutes, respectively, the LCM of 15 and 20 (which is 60) will tell us that the buses will depart together again after 60 minutes.

  2. Dividing and Grouping: HCF is useful in dividing objects or people into equal groups. For example, if we have 24 apples and 36 oranges, the HCF of 24 and 36 (which is 12) will tell us that we can make a maximum of 12 groups, each containing 2 apples and 3 oranges.

  3. Simplifying Fractions: HCF is also used to simplify fractions by dividing both the numerator and denominator by their HCF. For instance, the fraction 24/36 can be simplified by dividing both 24 and 36 by their HCF, which is 12, resulting in the simplified fraction 2/3.

In this comprehensive guide, we have embarked on a journey through the realms of LCM and HCF, unraveling their definitions, properties, and applications. We have witnessed how these concepts serve as the cornerstones of number relationships, providing a framework for solving a wide range of mathematical problems. By mastering the art of LCM and HCF, you have equipped yourself with valuable tools that extend beyond the realm of mathematics, empowering you to tackle real-world challenges with precision and confidence.

As you continue your mathematical explorations, remember that LCM and HCF are not isolated concepts; they are interconnected threads in the tapestry of numbers. By understanding their interplay and harnessing their power, you will unlock a deeper appreciation for the beauty and elegance of mathematics.

  • Least Common Multiple (LCM): The smallest positive integer divisible by all given numbers.
  • Highest Common Factor (HCF): The largest positive integer that divides all given numbers without a remainder.
  • Fundamental Formula: LCM (a, b) × HCF (a, b) = a × b
  • Euclidean Algorithm: A systematic method for finding the HCF of two numbers.
  • Real-World Applications: Scheduling, grouping, simplifying fractions.

With these keywords and concepts firmly in your grasp, you are well-equipped to conquer any numerical challenge that comes your way. Embrace the power of LCM and HCF, and let them guide you on your mathematical journey.