Find The Highest Common Factor (HCF) Of 150 And 405. Find The Highest Common Factor (HCF) Of 390 And 468. Find The Least Common Multiple (LCM) Of 60, 75, And 120. Find The Least Common Multiple (LCM) Of 30, 150, And 300.

by ADMIN 221 views

In the realm of mathematics, the concepts of Highest Common Factor (HCF) and Least Common Multiple (LCM) stand as fundamental pillars, playing a crucial role in various mathematical operations and problem-solving scenarios. Understanding these concepts is not just essential for academic pursuits but also for real-world applications, ranging from simplifying fractions to scheduling events. This comprehensive guide aims to demystify HCF and LCM, providing clear explanations, step-by-step methods, and practical examples to solidify your understanding. In this article, we will delve into the methods of finding the HCF and LCM of given numbers. Specifically, we will address the following problems:

  • Finding the HCF of 150 and 405
  • Finding the HCF of 390 and 468
  • Finding the LCM of 60, 75, and 120
  • Finding the LCM of 30, 150, and 300

Before we dive into solving these problems, let's first establish a clear understanding of what HCF and LCM mean and the different methods we can use to find them.

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers without leaving a remainder. In simpler terms, it's the biggest number that can perfectly divide all the given numbers. Finding the HCF is crucial in simplifying fractions, solving algebraic equations, and in various other mathematical contexts. There are several methods to determine the HCF, including the listing method, prime factorization method, and the division method. Each method offers a unique approach, and the choice of method often depends on the specific numbers involved and personal preference.

To truly grasp the concept of HCF, let's delve deeper into its significance and explore why it's such a valuable tool in mathematics. The HCF helps us simplify complex problems by breaking down numbers into their fundamental components. This simplification is particularly useful when dealing with fractions, where finding the HCF of the numerator and denominator allows us to reduce the fraction to its simplest form. For instance, if we have a fraction like 150/405, finding the HCF of 150 and 405 will enable us to simplify this fraction and express it in a more manageable form. Furthermore, the concept of HCF extends beyond simple arithmetic and finds applications in more advanced mathematical fields such as number theory and cryptography. Understanding HCF not only enhances our mathematical skills but also provides a foundation for tackling more complex problems.

  • Listing Method: This involves listing all the factors of each number and identifying the largest factor common to all. While straightforward, this method can be time-consuming for larger numbers.
  • Prime Factorization Method: This involves breaking down each number into its prime factors. The HCF is then the product of the common prime factors, each raised to the lowest power it appears in any of the factorizations. This method is generally more efficient for larger numbers.
  • Division Method: This method, also known as Euclid's algorithm, involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is zero. The last non-zero remainder is the HCF. This method is particularly useful for very large numbers.

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. In other words, it's the smallest number that all the given numbers can divide into without leaving a remainder. The LCM is essential in various mathematical operations, such as adding and subtracting fractions with different denominators, solving problems related to time and work, and in many other practical scenarios. Similar to HCF, there are several methods to calculate the LCM, including the listing method, prime factorization method, and the division method. Each method provides a different perspective on finding the LCM, and the choice of method depends on the specific numbers involved and the problem at hand.

The LCM is a cornerstone in various mathematical applications, and its understanding is crucial for solving a wide array of problems. One of the most common applications of LCM is in the addition and subtraction of fractions. When fractions have different denominators, we need to find the LCM of the denominators to obtain a common denominator, which allows us to perform the addition or subtraction. For example, if we need to add fractions with denominators 60, 75, and 120, finding the LCM of these numbers will give us the common denominator needed to proceed with the calculation. Beyond fractions, LCM is also used in problems involving cyclical events, such as determining when two events will occur simultaneously again, or in scheduling tasks that have different frequencies. A solid grasp of LCM not only simplifies mathematical calculations but also enhances our ability to solve real-world problems efficiently.

  • Listing Method: This involves listing multiples of each number until a common multiple is found. The smallest common multiple is the LCM. This method is suitable for smaller numbers.
  • Prime Factorization Method: This involves breaking down each number into its prime factors. The LCM is then the product of all prime factors, each raised to the highest power it appears in any of the factorizations. This method is generally more efficient for larger numbers.
  • Division Method: This method involves dividing the numbers by common prime factors until all the quotients are 1. The LCM is the product of the divisors and the final quotients. This method is particularly useful for finding the LCM of three or more numbers.

Let's begin by finding the HCF of 150 and 405. We will use the prime factorization method as it is efficient for these numbers.

  1. Prime Factorization of 150:
    • 150 = 2 × 75
    • 75 = 3 × 25
    • 25 = 5 × 5
    • So, 150 = 2 × 3 × 5 × 5 = 2 × 3 × 52
  2. Prime Factorization of 405:
    • 405 = 5 × 81
    • 81 = 3 × 27
    • 27 = 3 × 9
    • 9 = 3 × 3
    • So, 405 = 5 × 3 × 3 × 3 × 3 = 34 × 5
  3. Identify Common Prime Factors:
    • The common prime factors are 3 and 5.
  4. Determine the Lowest Powers:
    • The lowest power of 3 is 31
    • The lowest power of 5 is 51
  5. Calculate the HCF:
    • HCF (150, 405) = 31 × 51 = 3 × 5 = 15

Therefore, the HCF of 150 and 405 is 15. This means that 15 is the largest number that can divide both 150 and 405 without leaving a remainder. The prime factorization method is particularly effective because it breaks down each number into its fundamental building blocks, allowing us to easily identify common factors. By focusing on the common prime factors and their lowest powers, we can efficiently determine the HCF, even for larger numbers. This method not only provides the answer but also gives us a deeper understanding of the numbers themselves and their relationships. In the context of simplifying fractions, knowing that the HCF of 150 and 405 is 15 allows us to reduce the fraction 150/405 to its simplest form by dividing both the numerator and the denominator by 15, resulting in the simplified fraction 10/27.

Next, let's find the HCF of 390 and 468. We will again use the prime factorization method.

  1. Prime Factorization of 390:
    • 390 = 2 × 195
    • 195 = 3 × 65
    • 65 = 5 × 13
    • So, 390 = 2 × 3 × 5 × 13
  2. Prime Factorization of 468:
    • 468 = 2 × 234
    • 234 = 2 × 117
    • 117 = 3 × 39
    • 39 = 3 × 13
    • So, 468 = 2 × 2 × 3 × 3 × 13 = 22 × 32 × 13
  3. Identify Common Prime Factors:
    • The common prime factors are 2, 3, and 13.
  4. Determine the Lowest Powers:
    • The lowest power of 2 is 21
    • The lowest power of 3 is 31
    • The lowest power of 13 is 131
  5. Calculate the HCF:
    • HCF (390, 468) = 21 × 31 × 131 = 2 × 3 × 13 = 78

Thus, the HCF of 390 and 468 is 78. This means that 78 is the largest number that can divide both 390 and 468 without leaving a remainder. This example further illustrates the power and efficiency of the prime factorization method in determining the HCF. By breaking down each number into its prime factors, we can systematically identify the common factors and their lowest powers, ensuring an accurate calculation of the HCF. Understanding the prime factorization method not only helps in finding the HCF but also provides insights into the divisibility and relationships between numbers. In this case, knowing that the HCF of 390 and 468 is 78 can be useful in various mathematical contexts, such as simplifying ratios or solving equations involving these numbers.

Now, let's move on to finding the LCM of 60, 75, and 120. We will use the prime factorization method again.

  1. Prime Factorization of 60:
    • 60 = 2 × 30
    • 30 = 2 × 15
    • 15 = 3 × 5
    • So, 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
  2. Prime Factorization of 75:
    • 75 = 3 × 25
    • 25 = 5 × 5
    • So, 75 = 3 × 5 × 5 = 3 × 52
  3. Prime Factorization of 120:
    • 120 = 2 × 60
    • 60 = 2 × 30
    • 30 = 2 × 15
    • 15 = 3 × 5
    • So, 120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
  4. Identify All Prime Factors:
    • The prime factors are 2, 3, and 5.
  5. Determine the Highest Powers:
    • The highest power of 2 is 23
    • The highest power of 3 is 31
    • The highest power of 5 is 52
  6. Calculate the LCM:
    • LCM (60, 75, 120) = 23 × 31 × 52 = 8 × 3 × 25 = 600

Therefore, the LCM of 60, 75, and 120 is 600. This means that 600 is the smallest number that is divisible by 60, 75, and 120 without leaving a remainder. The prime factorization method shines in its ability to handle multiple numbers simultaneously, making it an efficient choice for finding the LCM of a set of numbers. By breaking each number down into its prime factors and then considering the highest power of each prime factor present, we can accurately calculate the LCM. This method provides a systematic approach that minimizes errors and ensures a clear understanding of the process. In practical applications, the LCM is essential when dealing with scenarios involving recurring events or cycles, such as scheduling tasks or aligning different time intervals. Knowing the LCM of 60, 75, and 120, for example, could be useful in coordinating events that occur at these intervals.

Finally, let's find the LCM of 30, 150, and 300. We will use the prime factorization method once more.

  1. Prime Factorization of 30:
    • 30 = 2 × 15
    • 15 = 3 × 5
    • So, 30 = 2 × 3 × 5
  2. Prime Factorization of 150:
    • 150 = 2 × 75
    • 75 = 3 × 25
    • 25 = 5 × 5
    • So, 150 = 2 × 3 × 5 × 5 = 2 × 3 × 52
  3. Prime Factorization of 300:
    • 300 = 2 × 150
    • 150 = 2 × 75
    • 75 = 3 × 25
    • 25 = 5 × 5
    • So, 300 = 2 × 2 × 3 × 5 × 5 = 22 × 3 × 52
  4. Identify All Prime Factors:
    • The prime factors are 2, 3, and 5.
  5. Determine the Highest Powers:
    • The highest power of 2 is 22
    • The highest power of 3 is 31
    • The highest power of 5 is 52
  6. Calculate the LCM:
    • LCM (30, 150, 300) = 22 × 31 × 52 = 4 × 3 × 25 = 300

Thus, the LCM of 30, 150, and 300 is 300. This means that 300 is the smallest number that is divisible by 30, 150, and 300 without leaving a remainder. This example further demonstrates the utility of the prime factorization method in determining the LCM, especially when dealing with numbers that have common factors. In this particular case, we observe that 300 is a multiple of both 30 and 150, which simplifies the process of finding the LCM. Recognizing such relationships between numbers can often streamline the calculation and provide a deeper understanding of their mathematical properties. The ability to efficiently find the LCM is crucial in various mathematical applications, from solving fractional equations to optimizing processes in real-world scenarios.

In conclusion, understanding the concepts of HCF and LCM is paramount in mathematics. We have explored the definitions, methods for calculation, and solved specific problems to illustrate these concepts. The prime factorization method has proven to be a versatile and efficient tool for finding both HCF and LCM. By mastering these concepts, you can tackle a wide range of mathematical problems with confidence and precision. Whether you are simplifying fractions, scheduling events, or solving complex equations, a solid understanding of HCF and LCM will undoubtedly prove invaluable. This guide has provided a comprehensive overview, equipping you with the knowledge and skills necessary to excel in this fundamental area of mathematics. Keep practicing and applying these concepts, and you'll find that the world of numbers becomes more accessible and understandable.