Calculate Sum Of HCF And LCM Of 35, 20, And 25 A Step-by-Step Guide

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In mathematics, understanding the concepts of Highest Common Factor (HCF) and Least Common Multiple (LCM) is crucial for solving various problems related to number theory. These concepts are fundamental in simplifying fractions, solving algebraic equations, and even in real-world applications like scheduling and resource allocation. In this article, we will delve into a step-by-step process to find the HCF and LCM of the numbers 35, 20, and 25, and then calculate their sum. This exercise will not only reinforce your understanding of these concepts but also demonstrate a practical application of these mathematical tools.

Understanding HCF and LCM

Before we jump into the calculation, let's first understand what HCF and LCM actually mean. 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 factor that the numbers share. For example, the HCF of 12 and 18 is 6, as 6 is the largest number that divides both 12 and 18 evenly. Finding the HCF is essential in various mathematical operations, such as simplifying fractions and solving equations. Understanding and accurately calculating the HCF allows for more efficient problem-solving and a deeper comprehension of numerical relationships.

On the other hand, the Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. It's the smallest number that each of the given numbers can divide into without leaving a remainder. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 divide into evenly. The LCM is particularly useful in situations where we need to find a common denominator for fractions or when dealing with cyclical events. By understanding the LCM, we can simplify complex mathematical problems and gain a stronger grasp of numerical patterns and relationships. Both HCF and LCM are foundational concepts in number theory and have widespread applications in various mathematical and real-world scenarios. Therefore, mastering these concepts is crucial for anyone looking to enhance their mathematical skills.

Step 1: Finding the HCF of 35, 20, and 25

To find the HCF of 35, 20, and 25, we can use the prime factorization method. This method involves breaking down each number into its prime factors, which are the prime numbers that multiply together to give the original number. Let's start by finding the prime factors of each number:

  • 35 = 5 x 7
  • 20 = 2 x 2 x 5
  • 25 = 5 x 5

Once we have the prime factors, we identify the common prime factors among the numbers. In this case, the only common prime factor among 35, 20, and 25 is 5. The HCF is the product of these common prime factors. Since 5 is the only common prime factor, the HCF of 35, 20, and 25 is 5. This means that 5 is the largest number that can divide 35, 20, and 25 without leaving a remainder. Understanding this process of finding the HCF is crucial as it forms the foundation for many other mathematical concepts and applications. It allows us to simplify fractions, solve equations, and even tackle real-world problems involving resource allocation and scheduling. The ability to quickly and accurately determine the HCF of a set of numbers is a valuable skill in both academic and practical contexts.

Step 2: Finding the LCM of 35, 20, and 25

Now, let's move on to finding the LCM of 35, 20, and 25. Similar to finding the HCF, we can use the prime factorization method. We already have the prime factors from the previous step:

  • 35 = 5 x 7
  • 20 = 2 x 2 x 5
  • 25 = 5 x 5

To find the LCM, we take the highest power of each prime factor that appears in any of the numbers. This ensures that the LCM is divisible by each of the original numbers. Let's break it down:

  • The highest power of 2 is 2^2 (from 20)
  • The highest power of 5 is 5^2 (from 25)
  • The highest power of 7 is 7^1 (from 35)

Now, we multiply these together: LCM = 2^2 x 5^2 x 7 = 4 x 25 x 7 = 700. Therefore, the LCM of 35, 20, and 25 is 700. This means that 700 is the smallest number that is divisible by 35, 20, and 25. Understanding the LCM is crucial in various mathematical applications, such as finding common denominators for fractions and solving problems involving cyclical events. For example, if three buses leave a station at different intervals, the LCM can help determine when they will all leave the station together again. Mastering the process of finding the LCM enhances your problem-solving skills and provides a deeper understanding of numerical relationships. It's a valuable tool in both theoretical mathematics and practical scenarios.

Step 3: Calculating the Sum of HCF and LCM

Finally, to find the sum of the HCF and LCM of 35, 20, and 25, we simply add the values we calculated in the previous steps. We found that the HCF is 5 and the LCM is 700. Therefore, the sum is:

Sum = HCF + LCM = 5 + 700 = 705

So, the sum of the HCF and LCM of 35, 20, and 25 is 705. This completes our calculation. This final step demonstrates the practical application of both HCF and LCM in solving a specific problem. By understanding how to calculate these values and then combine them, we gain a more comprehensive understanding of number theory. The ability to find the sum of HCF and LCM is not just a mathematical exercise; it reinforces the underlying concepts and their interrelation. It also highlights the importance of accuracy and attention to detail in mathematical calculations. This entire process, from finding the prime factors to calculating the final sum, showcases the logical and sequential nature of mathematical problem-solving.

Conclusion

In conclusion, finding the sum of the HCF and LCM of numbers is a fundamental mathematical exercise that reinforces key concepts in number theory. We successfully calculated the HCF of 35, 20, and 25 to be 5, and the LCM to be 700. Adding these together, we found the sum to be 705. This process not only provides a numerical answer but also enhances our understanding of prime factorization, common factors, and multiples. Mastering these concepts is crucial for success in various areas of mathematics and in real-world applications. The ability to efficiently and accurately determine the HCF and LCM allows for simplification of complex problems, whether in algebra, calculus, or even in practical scenarios like scheduling and resource management. Therefore, regular practice and a strong grasp of these fundamental concepts are essential for anyone looking to excel in mathematics. This exercise serves as a valuable example of how different mathematical principles can be combined to solve a specific problem, highlighting the interconnected nature of mathematical knowledge.