1. Two Numbers Have An LCM Of 60 And An HCF Of 4. If One Number Is 12, What Is The Other Number? 2. Two Numbers Have An HCF Of 6 And An LCM Of 120. If One Number Is 24, What Is The Other Number?

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In mathematics, the concepts of the Lowest Common Multiple (LCM) and the Highest Common Factor (HCF) are fundamental in number theory. These concepts are crucial for solving a variety of problems, particularly those involving fractions, ratios, and divisibility. This article aims to delve into practical applications of LCM and HCF by solving problems where one number and either the LCM or HCF are given, and we need to find the other number. Understanding these methods not only enhances problem-solving skills but also provides a deeper insight into the relationships between numbers.

In the realm of number theory, finding a number given the LCM (Lowest Common Multiple), HCF (Highest Common Factor), and another number is a classic problem that elegantly demonstrates the relationship between these fundamental concepts. The problem we'll tackle here is: The LCM of two numbers is 60, and their HCF is 4. If one of the numbers is 12, calculate the other number. To effectively solve this, we will use a core principle that links the LCM and HCF of two numbers with the product of the numbers themselves. This principle states that the product of two numbers is equal to the product of their LCM and HCF. Understanding this relationship is vital not only for solving this specific problem but also for grasping broader concepts in number theory. Before diving into the solution, it's beneficial to understand what LCM and HCF represent. The LCM of two numbers is the smallest multiple that is divisible by both numbers. The HCF, on the other hand, is the largest number that divides both numbers without leaving a remainder. These concepts are not just theoretical; they have practical applications in various fields of mathematics and real-world scenarios, such as scheduling, dividing quantities, and simplifying fractions. The ability to calculate these values and understand their relationships can significantly enhance one's mathematical toolkit. Now, let's apply this principle to our problem. We know the LCM is 60, the HCF is 4, and one of the numbers is 12. We are looking to find the other number, which we'll call 'x'. Using the principle mentioned earlier, we can set up the equation: 12 * x = 60 * 4. This equation forms the basis of our solution, and by solving it, we can find the value of 'x'. The equation is straightforward but powerful, illustrating how fundamental mathematical principles can simplify seemingly complex problems. Solving such problems not only hones our calculation skills but also enhances our logical thinking and ability to apply mathematical concepts in diverse situations. The subsequent steps involve simple arithmetic, but it's the understanding of the initial principle that guides us to the solution. By methodically breaking down the problem and applying the relevant mathematical concepts, we can arrive at the correct answer and reinforce our understanding of number theory.

Solution:

  1. Understand the relationship: The fundamental relationship we use here is that the product of two numbers is equal to the product of their LCM and HCF.

  2. Set up the equation: Let the other number be x. So, 12 * x = LCM * HCF, which means 12 * x = 60 * 4.

  3. Solve for x:

    • 12x = 240
    • x = 240 / 12
    • x = 20

Therefore, the other number is 20. This problem underscores the importance of understanding and applying fundamental mathematical principles. The relationship between LCM, HCF, and the numbers themselves is a powerful tool for solving a variety of problems in number theory. By knowing this relationship, you can easily find one of the numbers if the other number, the LCM, and the HCF are known. Furthermore, this type of problem highlights the practical application of mathematical concepts in real-world scenarios. Understanding LCM and HCF is not just an academic exercise; it has implications in various fields, from computer science to engineering. In computer science, for instance, LCM and HCF are used in scheduling algorithms and data encryption. In engineering, they can be used in designing systems where components need to operate in sync or where resources need to be divided efficiently. The key takeaway from this problem is the ability to recognize and apply the relevant mathematical principles. It's not just about memorizing formulas but about understanding the underlying concepts and how they relate to each other. This approach is crucial for developing problem-solving skills that are transferable to other areas of mathematics and beyond. As we move on to the next problem, we will see another application of these concepts, further reinforcing the importance of a strong foundation in number theory. Remember, mathematics is a cumulative subject, where each concept builds upon the previous ones. Therefore, a thorough understanding of the basics is essential for tackling more complex problems.

In mathematics, particularly in number theory, the concept of finding a number given the Highest Common Factor (HCF), Lowest Common Multiple (LCM), and another number is a crucial skill. This type of problem often appears in various mathematical contexts, from basic arithmetic to more advanced algebraic applications. The problem at hand is as follows: The HCF of two numbers is 6, and their LCM is 120. If one of the numbers is 24, calculate the other number. This problem not only tests our understanding of HCF and LCM but also our ability to apply the relationship between these concepts and the numbers themselves. Before we delve into the solution, it's essential to revisit the definitions of HCF and LCM. The HCF of two numbers is the largest number that divides both numbers without leaving a remainder. In contrast, the LCM is the smallest multiple that is divisible by both numbers. These concepts are fundamental in simplifying fractions, understanding divisibility rules, and solving problems involving ratios and proportions. The relationship between HCF, LCM, and the two numbers is a cornerstone of number theory. It states that the product of two numbers is equal to the product of their HCF and LCM. This principle is not just a formula to be memorized; it's a powerful tool that allows us to solve a variety of problems efficiently. Understanding the derivation of this relationship can further enhance our grasp of the concepts involved. Now, let's apply this principle to the problem at hand. We are given that the HCF of two numbers is 6, their LCM is 120, and one of the numbers is 24. Our task is to find the other number. To do this, we will use the relationship we just discussed. Let the other number be y. According to the principle, we can set up the equation: 24 * y = 6 * 120. This equation is the key to solving the problem. It allows us to express the unknown number in terms of known quantities, making the problem solvable through simple algebraic manipulation. The ability to set up such equations is a critical skill in mathematics. It demonstrates not only an understanding of the underlying concepts but also the ability to translate a word problem into a mathematical expression. As we proceed to solve the equation, we will see how the fundamental principle of HCF and LCM helps us arrive at the solution. This problem is a testament to the power of mathematical principles in simplifying complex situations. By breaking down the problem into its components and applying the relevant concepts, we can find the answer and deepen our understanding of number theory.

Solution:

  1. Apply the relationship: Again, we use the same relationship: the product of two numbers is equal to the product of their LCM and HCF.

  2. Set up the equation: Let the other number be y. So, 24 * y = LCM * HCF, which means 24 * y = 120 * 6.

  3. Solve for y:

    • 24y = 720
    • y = 720 / 24
    • y = 30

Thus, the other number is 30. This solution reinforces the importance of understanding the relationship between HCF, LCM, and the numbers involved. The ability to apply this relationship effectively allows for the solution of various problems in number theory. Moreover, this type of problem highlights the interconnectedness of mathematical concepts. The HCF and LCM are not just isolated topics; they are linked to other areas of mathematics, such as divisibility, prime factorization, and modular arithmetic. A solid understanding of these concepts can provide a foundation for more advanced mathematical studies. Furthermore, the problem-solving process involved in this question is a valuable skill in itself. It requires the ability to analyze a problem, identify the relevant information, and apply the appropriate mathematical principles. These are skills that are not only useful in mathematics but also in other fields of study and in real-world situations. The ability to break down a complex problem into smaller, manageable parts and then solve each part systematically is a hallmark of effective problem-solving. The problem also illustrates the power of mathematical notation and equations in representing relationships and solving problems. The equation 24 * y = 120 * 6 is a concise and precise way of expressing the relationship between the numbers, their HCF, and their LCM. This ability to translate word problems into mathematical equations is a crucial skill for anyone studying mathematics or science. In conclusion, this problem serves as a valuable exercise in understanding and applying the concepts of HCF and LCM. It reinforces the importance of mathematical principles and the power of problem-solving skills in mathematics and beyond. As we have seen, by understanding the underlying concepts and applying them systematically, we can solve seemingly complex problems with ease.

In conclusion, both problems effectively illustrate the practical application of LCM and HCF in finding unknown numbers. The core principle that the product of two numbers equals the product of their LCM and HCF is a powerful tool for solving these types of problems. Mastering these concepts not only improves mathematical proficiency but also enhances problem-solving skills applicable in various real-world scenarios. These examples underscore the importance of a solid foundation in number theory for tackling more complex mathematical challenges. By understanding and applying these principles, one can approach similar problems with confidence and accuracy. The relationship between numbers, their LCM, and their HCF is a fundamental aspect of mathematics, and its understanding opens doors to a deeper appreciation of mathematical concepts and their applications.