Finding HCF By Long Division And Solving Remainder Problems

Jul 16, 2025 by ADMIN 60 views

Finding the Highest Common Factor (HCF) by Long Division Method and Solving Remainder Problems

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. The long division method is a systematic way to find the HCF, especially useful for larger numbers. This article will guide you through finding the HCF using the long division method with several examples and then delve into solving problems involving remainders.

(a) HCF of 2025 and 5184

To find the HCF of 2025 and 5184 using the long division method, we follow these steps:

Divide the larger number (5184) by the smaller number (2025).
If the remainder is not zero, divide the divisor (2025) by the remainder.
Repeat the process until the remainder is zero. The last non-zero divisor is the HCF.

Let’s apply this:

Step 1: Divide 5184 by 2025.
```
5184 = 2025 * 2 + 1134
```
The remainder is 1134.
Step 2: Divide 2025 by 1134.
```
2025 = 1134 * 1 + 891
```
The remainder is 891.
Step 3: Divide 1134 by 891.
```
1134 = 891 * 1 + 243
```
The remainder is 243.
Step 4: Divide 891 by 243.
```
891 = 243 * 3 + 162
```
The remainder is 162.
Step 5: Divide 243 by 162.
```
243 = 162 * 1 + 81
```
The remainder is 81.
Step 6: Divide 162 by 81.
```
162 = 81 * 2 + 0
```
The remainder is 0.

Therefore, the HCF of 2025 and 5184 is 81.

Understanding the long division method is crucial for solving such problems. It provides a structured approach to breaking down the numbers until the common factor is revealed. The process involves successive divisions, where the remainder from each step becomes the new divisor in the next step. This iterative approach ensures that we find the largest number that can divide both original numbers without leaving a remainder.

Moreover, the long division method is not just a mechanical process; it's a reflection of the fundamental principles of number theory. By repeatedly dividing and finding remainders, we are essentially peeling away layers of factors until we reach the core common factor. This method is particularly useful when dealing with large numbers where factorization might be cumbersome. The efficiency of the long division method lies in its ability to quickly narrow down the possibilities and converge on the HCF, making it an indispensable tool in mathematical problem-solving.

(b) HCF of 8064 and 4410

Let’s find the HCF of 8064 and 4410 using the same long division method.

Step 1: Divide 8064 by 4410.
```
8064 = 4410 * 1 + 3654
```
The remainder is 3654.
Step 2: Divide 4410 by 3654.
```
4410 = 3654 * 1 + 756
```
The remainder is 756.
Step 3: Divide 3654 by 756.
```
3654 = 756 * 4 + 630
```
The remainder is 630.
Step 4: Divide 756 by 630.
```
756 = 630 * 1 + 126
```
The remainder is 126.
Step 5: Divide 630 by 126.
```
630 = 126 * 5 + 0
```
The remainder is 0.

Therefore, the HCF of 8064 and 4410 is 126.

Understanding the application of the Euclidean algorithm, which underlies the long division method, is essential here. The Euclidean algorithm is an efficient method for computing the HCF of two numbers, and it's based on the principle that the HCF of two numbers also divides their difference. In each step of the long division, we are effectively reducing the problem to finding the HCF of smaller numbers, which ultimately leads us to the solution. The beauty of this method lies in its simplicity and its guaranteed convergence to the HCF, regardless of the size of the numbers involved. Furthermore, the Euclidean algorithm has numerous applications in cryptography and computer science, highlighting its significance beyond basic arithmetic.

By mastering the long division method, students gain a deeper appreciation for the properties of numbers and the relationships between them. This understanding not only helps in solving HCF problems but also lays a strong foundation for more advanced mathematical concepts. The ability to systematically break down a problem and apply a proven method is a valuable skill that extends beyond the realm of mathematics and into various aspects of problem-solving in general.

(c) HCF of 264, 840, and 384

To find the HCF of three numbers, we first find the HCF of any two numbers, and then find the HCF of the result with the third number.

Find the HCF of 264 and 840.
- Step 1: Divide 840 by 264.
```
840 = 264 * 3 + 48
```
  The remainder is 48.
- Step 2: Divide 264 by 48.
```
264 = 48 * 5 + 24
```
  The remainder is 24.
- Step 3: Divide 48 by 24.
```
48 = 24 * 2 + 0
```
  The remainder is 0.
So, the HCF of 264 and 840 is 24.
Now, find the HCF of 24 and 384.
- Step 1: Divide 384 by 24.
```
384 = 24 * 16 + 0
```
  The remainder is 0.
Therefore, the HCF of 264, 840, and 384 is 24.

When dealing with the HCF of multiple numbers, the concept of pairwise HCF becomes important. This approach involves finding the HCF of two numbers at a time and then using that result to find the HCF with the next number. This method is efficient and ensures that we find the largest number that divides all the given numbers without leaving a remainder. The key is to break down the problem into smaller, manageable steps, making the overall process less daunting and more straightforward.

Furthermore, understanding the properties of HCF in relation to multiple numbers is crucial for various applications, such as simplifying fractions and solving problems in number theory. The HCF represents the common ground among the numbers, the largest piece that fits perfectly into each of them. This concept is not only useful in mathematics but also in real-world scenarios, such as resource allocation and scheduling, where finding the common factor helps in optimizing the process.

The ability to extend the long division method to multiple numbers demonstrates a deeper understanding of the underlying mathematical principles. It showcases the versatility of the method and its applicability in different contexts. This skill is essential for students as they progress in their mathematical journey and encounter more complex problems involving number theory and algebra.

(d) HCF of 625, 3125, and 15625

Let’s find the HCF of 625, 3125, and 15625 using the same approach.

Find the HCF of 625 and 3125.
- Step 1: Divide 3125 by 625.
```
3125 = 625 * 5 + 0
```
  The remainder is 0.
So, the HCF of 625 and 3125 is 625.
Now, find the HCF of 625 and 15625.
- Step 1: Divide 15625 by 625.
```
15625 = 625 * 25 + 0
```
  The remainder is 0.
Therefore, the HCF of 625, 3125, and 15625 is 625.

In this example, we see that 3125 is a multiple of 625, and 15625 is a multiple of both 625 and 3125. This makes the process of finding the HCF straightforward. Recognizing these relationships can significantly simplify the calculations and save time. The divisibility rules play a crucial role here, allowing us to quickly identify common factors and simplify the problem.

Understanding the concept of multiples and factors is fundamental to finding the HCF. When one number is a multiple of another, the smaller number is the HCF. This insight is particularly useful in cases like this, where the numbers are clearly related by a common factor. The ability to identify these relationships showcases a strong grasp of number theory and can lead to efficient problem-solving strategies.

Moreover, this example illustrates the importance of observation and pattern recognition in mathematics. By noticing the relationships between the numbers, we can avoid unnecessary calculations and arrive at the solution more quickly. This skill is not only valuable in finding the HCF but also in various other mathematical contexts. It encourages a more intuitive approach to problem-solving, where understanding the underlying structure of the problem is as important as applying the correct method.

Now, let's tackle the problem of finding the greatest number that divides 28, 41, and 66, leaving remainders of 4, 5, and 6, respectively. This type of problem involves a slight twist on the HCF concept, requiring us to first adjust the numbers based on the remainders.

Problem: Find the greatest number that divides 28, 41, and 66 leaving remainders 4, 5, and 6 respectively.

To solve this, we first subtract the respective remainders from the numbers:

28 - 4 = 24
41 - 5 = 36
66 - 6 = 60

Now, we need to find the HCF of 24, 36, and 60. Let’s use the long division method or prime factorization.

Find the HCF of 24 and 36.
- Prime factorization of 24: 2^3 * 3
- Prime factorization of 36: 2^2 * 3^2
- HCF of 24 and 36: 2^2 * 3 = 12
Now, find the HCF of 12 and 60.
- Prime factorization of 12: 2^2 * 3
- Prime factorization of 60: 2^2 * 3 * 5
- HCF of 12 and 60: 2^2 * 3 = 12

Therefore, the greatest number that divides 28, 41, and 66, leaving remainders 4, 5, and 6 respectively, is 12.

This problem highlights the importance of understanding how remainders affect divisibility. By subtracting the remainders, we transform the problem into a standard HCF problem. This technique is a common strategy in number theory and is essential for solving a variety of problems involving divisors and remainders. The ability to manipulate the given information and reframe the problem in a more familiar form is a key problem-solving skill.

Furthermore, this example demonstrates the versatility of the HCF concept. It's not just about finding the largest common factor of a set of numbers; it's also about understanding how to apply this concept in different contexts. Problems involving remainders require a deeper understanding of the relationship between divisors, dividends, and remainders. By mastering these concepts, students can develop a more comprehensive understanding of number theory and its applications.

The choice between the long division method and prime factorization depends on the specific problem and personal preference. Both methods are effective for finding the HCF, but one might be more efficient than the other in certain cases. Understanding the strengths and weaknesses of each method allows students to choose the most appropriate approach for a given problem. This adaptability is a valuable asset in mathematical problem-solving.

In conclusion, finding the HCF using the long division method is a fundamental skill in mathematics. By understanding the process and applying it to various problems, including those involving remainders, students can develop a strong foundation in number theory and problem-solving. The examples provided in this article serve as a comprehensive guide to mastering this important concept.