How To Find The Highest Common Factor (HCF) A Step-by-Step Guide

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The highest common factor (HCF), also known as the greatest common divisor (GCD), is a fundamental concept in number theory. It represents the largest positive integer that divides two or more integers without leaving a remainder. Understanding HCF is crucial for simplifying fractions, solving mathematical problems, and various real-world applications. This article will guide you through finding the HCF of different pairs and groups of numbers, providing a step-by-step approach to mastering this essential mathematical skill.

The highest common factor (HCF), or greatest common divisor (GCD), is a cornerstone of number theory, pivotal in simplifying fractions, tackling mathematical problems, and real-world applications. To truly grasp HCF, we must delve into its essence, methods of calculation, and practical significance. This article serves as a comprehensive guide, providing a step-by-step approach to mastering HCF calculations for both pairs and groups of numbers. Understanding HCF is like unlocking a secret code that simplifies complex mathematical problems. For instance, when dealing with fractions, finding the HCF of the numerator and denominator allows us to reduce the fraction to its simplest form, making it easier to work with. Moreover, HCF plays a crucial role in various fields, from computer science to cryptography, highlighting its versatility and importance in the world around us. In essence, mastering HCF is not just about crunching numbers; it's about developing a foundational skill that enhances your mathematical prowess and problem-solving abilities across diverse domains. So, let's embark on this journey of understanding HCF, where we'll explore its intricacies, practice its application, and uncover its significance in the broader context of mathematics and beyond. This knowledge will empower you to approach numerical challenges with confidence and clarity, making you a more proficient problem solver in both academic and real-world scenarios.

There are primarily two methods for finding the HCF:

  1. Prime Factorization Method: This method involves expressing each number as a product of its prime factors and then identifying the common prime factors with the lowest powers.
  2. Division Method: This method involves repeatedly dividing the larger number by the smaller number until the remainder is zero. The last non-zero divisor is the HCF.

The prime factorization method and the division method stand as the two primary techniques for determining the HCF, each offering a unique pathway to the solution. The prime factorization method is akin to dissecting a number into its fundamental building blocks. By expressing each number as a product of its prime factors, we unveil the common elements that contribute to their shared divisibility. This method is particularly insightful when dealing with smaller numbers or when a clear understanding of the numbers' composition is desired. Identifying the common prime factors with the lowest powers allows us to construct the HCF, ensuring it's the largest number that divides both original numbers without leaving a remainder. On the other hand, the division method, also known as the Euclidean algorithm, offers a more iterative approach. It involves a systematic process of repeatedly dividing the larger number by the smaller number, then using the remainder as the new divisor, until a remainder of zero is achieved. The last non-zero divisor in this process reveals the HCF. This method is especially efficient when dealing with larger numbers, as it avoids the need to find all prime factors. Choosing between these two methods often depends on the specific numbers involved and personal preference. However, mastering both methods equips you with a versatile toolkit for tackling HCF problems, enhancing your problem-solving skills and mathematical agility. So, whether you prefer the analytical dissection of prime factorization or the iterative elegance of the division method, understanding both techniques will undoubtedly strengthen your grasp of HCF and its applications.

Let's apply these methods to find the HCF of the given pairs of numbers.

(a) 8 and 12

Prime Factorization Method

  • Prime factors of 8: 2 x 2 x 2 = 2³
  • Prime factors of 12: 2 x 2 x 3 = 2² x 3
  • Common prime factors: 2²
  • HCF (8, 12) = 2² = 4

Division Method

  • 12 ÷ 8 = 1 (remainder 4)
  • 8 ÷ 4 = 2 (remainder 0)
  • HCF (8, 12) = 4

Finding the highest common factor (HCF) of 8 and 12 provides a practical example of how the prime factorization and division methods converge to the same solution. When we break down 8 and 12 into their prime factors, we unveil the underlying structure of these numbers. The prime factorization of 8 is 2 x 2 x 2, or 2³, indicating that 2 is the sole prime building block of 8, repeated three times. Similarly, the prime factorization of 12 is 2 x 2 x 3, or 2² x 3, revealing that 12 is constructed from two 2s and a 3. By identifying the common prime factors, which in this case are two 2s (2²), we can construct the HCF. This method not only gives us the answer but also offers a deeper understanding of the numbers' composition. On the other hand, the division method presents a more algorithmic approach. We begin by dividing the larger number (12) by the smaller number (8), obtaining a quotient of 1 and a remainder of 4. This remainder then becomes the new divisor, and we divide the previous divisor (8) by it. The process continues until we reach a remainder of 0. The last non-zero divisor, which is 4 in this case, is the HCF. This method is particularly efficient as it systematically reduces the problem to smaller numbers, eventually revealing the HCF. Both methods, though different in their approach, highlight the fundamental principle of HCF: finding the largest number that divides both given numbers without leaving a remainder. The fact that both methods lead to the same answer reinforces the robustness of the concept and provides confidence in our calculations. Understanding these methods not only equips us with practical tools but also enhances our mathematical intuition, allowing us to tackle more complex problems with ease.

(b) 18 and 27

Prime Factorization Method

  • Prime factors of 18: 2 x 3 x 3 = 2 x 3²
  • Prime factors of 27: 3 x 3 x 3 = 3³
  • Common prime factors: 3²
  • HCF (18, 27) = 3² = 9

Division Method

  • 27 ÷ 18 = 1 (remainder 9)
  • 18 ÷ 9 = 2 (remainder 0)
  • HCF (18, 27) = 9

In the quest to determine the highest common factor (HCF) of 18 and 27, we once again encounter the elegance and consistency of both the prime factorization and division methods. Let's delve into the prime factorization method first. By dissecting 18 and 27 into their prime constituents, we gain a clearer understanding of their divisibility. The prime factorization of 18 is 2 x 3 x 3, which can be succinctly expressed as 2 x 3². This reveals that 18 is composed of the prime numbers 2 and 3, with 3 appearing twice. Similarly, the prime factorization of 27 is 3 x 3 x 3, or 3³, indicating that 27 is exclusively built from the prime number 3, repeated three times. To find the HCF, we identify the common prime factors, which in this case are the two instances of 3 (3²). Multiplying these common factors together yields 3², which equals 9, thus establishing the HCF of 18 and 27. Now, let's explore the division method, also known as the Euclidean algorithm, a systematic approach that iteratively reduces the problem until the HCF is revealed. We begin by dividing the larger number (27) by the smaller number (18), resulting in a quotient of 1 and a remainder of 9. This remainder then becomes the new divisor, and we divide the previous divisor (18) by it. This yields a quotient of 2 and a remainder of 0. The process halts when we reach a remainder of 0, and the last non-zero divisor, which is 9, becomes the HCF. Once again, both methods converge on the same answer, reinforcing the reliability and versatility of these techniques. Whether you prefer the analytical precision of prime factorization or the algorithmic efficiency of the division method, mastering both empowers you to tackle HCF problems with confidence and clarity. The journey of finding the HCF of 18 and 27 serves as a testament to the beauty and consistency of mathematical principles, highlighting the power of understanding the underlying structure of numbers and the diverse approaches available to solve numerical challenges.

(c) 21 and 84

Prime Factorization Method

  • Prime factors of 21: 3 x 7
  • Prime factors of 84: 2 x 2 x 3 x 7 = 2² x 3 x 7
  • Common prime factors: 3 x 7
  • HCF (21, 84) = 3 x 7 = 21

Division Method

  • 84 ÷ 21 = 4 (remainder 0)
  • HCF (21, 84) = 21

Determining the highest common factor (HCF) of 21 and 84 presents an intriguing case study that further illustrates the efficacy of both the prime factorization and division methods. Let's embark on this exploration, beginning with the prime factorization method. By deconstructing 21 and 84 into their fundamental prime constituents, we unlock the secrets of their divisibility. The prime factorization of 21 is simply 3 x 7, indicating that 21 is composed of the prime numbers 3 and 7. On the other hand, the prime factorization of 84 is 2 x 2 x 3 x 7, which can be elegantly expressed as 2² x 3 x 7. This reveals that 84 is built from the prime numbers 2, 3, and 7, with 2 appearing twice. To unearth the HCF, we identify the common prime factors shared by both numbers, which are 3 and 7. Multiplying these common factors together, we obtain 3 x 7, which equals 21, thus affirming the HCF of 21 and 84. Now, let's shift our focus to the division method, also known as the Euclidean algorithm, a systematic process that iteratively narrows down the possibilities until the HCF emerges. We commence by dividing the larger number (84) by the smaller number (21), resulting in a quotient of 4 and a remainder of 0. This is a unique scenario where the division yields a remainder of 0 in the first step. In such cases, the smaller number, which is 21, is directly declared as the HCF. This outcome underscores the efficiency of the division method, particularly when one number is a multiple of the other. Once again, both methods converge on the same answer, reinforcing the reliability and versatility of these techniques. Whether you are drawn to the analytical precision of prime factorization or the algorithmic efficiency of the division method, mastering both equips you to confidently navigate HCF challenges. The journey of finding the HCF of 21 and 84 serves as a testament to the power of mathematical tools in unraveling numerical relationships and the beauty of consistent results achieved through diverse approaches.

(d) 72 and 90

Prime Factorization Method

  • Prime factors of 72: 2 x 2 x 2 x 3 x 3 = 2³ x 3²
  • Prime factors of 90: 2 x 3 x 3 x 5 = 2 x 3² x 5
  • Common prime factors: 2 x 3²
  • HCF (72, 90) = 2 x 3² = 18

Division Method

  • 90 ÷ 72 = 1 (remainder 18)
  • 72 ÷ 18 = 4 (remainder 0)
  • HCF (72, 90) = 18

The endeavor to find the highest common factor (HCF) of 72 and 90 presents an opportunity to further solidify our understanding of the prime factorization and division methods. Let's embark on this mathematical journey, starting with the prime factorization method. By dissecting 72 and 90 into their fundamental prime constituents, we gain valuable insights into their divisibility properties. The prime factorization of 72 is 2 x 2 x 2 x 3 x 3, which can be elegantly expressed as 2³ x 3². This reveals that 72 is composed of the prime numbers 2 and 3, with 2 appearing three times and 3 appearing twice. Similarly, the prime factorization of 90 is 2 x 3 x 3 x 5, or 2 x 3² x 5, indicating that 90 is built from the prime numbers 2, 3, and 5, with 3 appearing twice. To determine the HCF, we identify the common prime factors shared by both numbers, which are 2 and two instances of 3 (3²). Multiplying these common factors together, we obtain 2 x 3², which equals 18, thus establishing the HCF of 72 and 90. Now, let's turn our attention to the division method, also known as the Euclidean algorithm, a systematic approach that iteratively refines the problem until the HCF is revealed. We begin by dividing the larger number (90) by the smaller number (72), resulting in a quotient of 1 and a remainder of 18. This remainder then becomes the new divisor, and we divide the previous divisor (72) by it. This yields a quotient of 4 and a remainder of 0. The process concludes when we reach a remainder of 0, and the last non-zero divisor, which is 18, becomes the HCF. Once again, both methods converge on the same answer, underscoring the reliability and versatility of these techniques. Whether you gravitate towards the analytical precision of prime factorization or the algorithmic efficiency of the division method, mastering both equips you to confidently tackle HCF challenges. The exploration of finding the HCF of 72 and 90 serves as a testament to the power of mathematical tools in unraveling numerical relationships and the consistency of results achieved through diverse methodologies.

(e) 74 and 99

Prime Factorization Method

  • Prime factors of 74: 2 x 37
  • Prime factors of 99: 3 x 3 x 11 = 3² x 11
  • Common prime factors: None
  • HCF (74, 99) = 1

Division Method

  • 99 ÷ 74 = 1 (remainder 25)
  • 74 ÷ 25 = 2 (remainder 24)
  • 25 ÷ 24 = 1 (remainder 1)
  • 24 ÷ 1 = 24 (remainder 0)
  • HCF (74, 99) = 1

The task of determining the highest common factor (HCF) of 74 and 99 presents a unique scenario that highlights the importance of recognizing numbers that share no common factors other than 1. Let's delve into this exploration, beginning with the prime factorization method. By dissecting 74 and 99 into their fundamental prime constituents, we aim to uncover their divisibility properties. The prime factorization of 74 is 2 x 37, indicating that 74 is composed of the prime numbers 2 and 37. On the other hand, the prime factorization of 99 is 3 x 3 x 11, which can be expressed as 3² x 11. This reveals that 99 is built from the prime numbers 3 and 11. Upon comparing the prime factors of 74 and 99, we observe that they share no common prime factors. In such cases, where numbers have no prime factors in common, their HCF is 1. This signifies that 1 is the largest number that can divide both 74 and 99 without leaving a remainder. Now, let's examine the division method, also known as the Euclidean algorithm, a systematic approach that iteratively refines the problem until the HCF is revealed. We begin by dividing the larger number (99) by the smaller number (74), resulting in a quotient of 1 and a remainder of 25. This remainder then becomes the new divisor, and we divide the previous divisor (74) by it, yielding a quotient of 2 and a remainder of 24. We continue this process, dividing 25 by 24, which gives a quotient of 1 and a remainder of 1. Finally, we divide 24 by 1, resulting in a quotient of 24 and a remainder of 0. The process concludes when we reach a remainder of 0, and the last non-zero divisor, which is 1, becomes the HCF. The result obtained through both methods, an HCF of 1, underscores the concept of relatively prime numbers, which are numbers that share no common factors other than 1. The exploration of finding the HCF of 74 and 99 serves as a valuable reminder that not all numbers share common factors, and in such instances, the HCF is simply 1. This understanding reinforces the nuances of number theory and the diverse relationships that can exist between numbers.

(f) 120 and 225

Prime Factorization Method

  • Prime factors of 120: 2 x 2 x 2 x 3 x 5 = 2³ x 3 x 5
  • Prime factors of 225: 3 x 3 x 5 x 5 = 3² x 5²
  • Common prime factors: 3 x 5
  • HCF (120, 225) = 3 x 5 = 15

Division Method

  • 225 ÷ 120 = 1 (remainder 105)
  • 120 ÷ 105 = 1 (remainder 15)
  • 105 ÷ 15 = 7 (remainder 0)
  • HCF (120, 225) = 15

Finding the highest common factor (HCF) of 120 and 225 provides an opportunity to apply our understanding of prime factorization and the division method to larger numbers. Let's begin by exploring the prime factorization method. By breaking down 120 and 225 into their prime factors, we can identify their common divisors. The prime factorization of 120 is 2 x 2 x 2 x 3 x 5, which can be written as 2³ x 3 x 5. This tells us that 120 is composed of the prime numbers 2, 3, and 5. The prime factorization of 225 is 3 x 3 x 5 x 5, or 3² x 5². This shows that 225 is composed of the prime numbers 3 and 5. To find the HCF, we look for the common prime factors with the lowest powers. Both numbers share the prime factors 3 and 5. The lowest power of 3 present in both factorizations is 3¹, and the lowest power of 5 is 5¹. Therefore, the HCF is the product of these common prime factors: 3 x 5 = 15. Now, let's use the division method, also known as the Euclidean algorithm, to verify our result. We start by dividing the larger number (225) by the smaller number (120). 225 divided by 120 gives a quotient of 1 and a remainder of 105. Next, we divide 120 by the remainder 105, which gives a quotient of 1 and a remainder of 15. We continue this process by dividing 105 by the remainder 15. 105 divided by 15 gives a quotient of 7 and a remainder of 0. Since the remainder is 0, the last non-zero divisor, which is 15, is the HCF. Both the prime factorization method and the division method give us the same HCF of 15 for 120 and 225. This consistency reinforces the reliability of both methods and provides confidence in our calculations. Working with larger numbers like 120 and 225 demonstrates the practical application of these HCF-finding techniques and highlights their importance in simplifying mathematical problems.

(g) 108 and 240

Prime Factorization Method

  • Prime factors of 108: 2 x 2 x 3 x 3 x 3 = 2² x 3³
  • Prime factors of 240: 2 x 2 x 2 x 2 x 3 x 5 = 2⁴ x 3 x 5
  • Common prime factors: 2² x 3
  • HCF (108, 240) = 2² x 3 = 12

Division Method

  • 240 ÷ 108 = 2 (remainder 24)
  • 108 ÷ 24 = 4 (remainder 12)
  • 24 ÷ 12 = 2 (remainder 0)
  • HCF (108, 240) = 12

The process of determining the highest common factor (HCF) of 108 and 240 allows us to further apply our knowledge of prime factorization and the division method to larger, more complex numbers. Let's start with the prime factorization method. By expressing 108 and 240 as products of their prime factors, we can identify the common divisors. The prime factorization of 108 is 2 x 2 x 3 x 3 x 3, which can be written as 2² x 3³. This means that 108 is composed of the prime numbers 2 and 3. The prime factorization of 240 is 2 x 2 x 2 x 2 x 3 x 5, or 2⁴ x 3 x 5. This shows that 240 is composed of the prime numbers 2, 3, and 5. To find the HCF, we identify the common prime factors with the lowest powers. Both numbers share the prime factors 2 and 3. The lowest power of 2 present in both factorizations is 2², and the lowest power of 3 is 3¹. Therefore, the HCF is the product of these common prime factors: 2² x 3 = 4 x 3 = 12. Now, let's use the division method, also known as the Euclidean algorithm, to confirm our result. We begin by dividing the larger number (240) by the smaller number (108). 240 divided by 108 gives a quotient of 2 and a remainder of 24. Next, we divide 108 by the remainder 24, which gives a quotient of 4 and a remainder of 12. We continue this process by dividing 24 by the remainder 12. 24 divided by 12 gives a quotient of 2 and a remainder of 0. Since the remainder is 0, the last non-zero divisor, which is 12, is the HCF. The HCF of 108 and 240, as determined by both the prime factorization method and the division method, is 12. This consistency reinforces the effectiveness of these methods and demonstrates their applicability to a wide range of numbers. Working with numbers like 108 and 240 showcases the utility of these HCF-finding techniques in simplifying mathematical problems and gaining a deeper understanding of number relationships.

(h) 231 and 396

Prime Factorization Method

  • Prime factors of 231: 3 x 7 x 11
  • Prime factors of 396: 2 x 2 x 3 x 3 x 11 = 2² x 3² x 11
  • Common prime factors: 3 x 11
  • HCF (231, 396) = 3 x 11 = 33

Division Method

  • 396 ÷ 231 = 1 (remainder 165)
  • 231 ÷ 165 = 1 (remainder 66)
  • 165 ÷ 66 = 2 (remainder 33)
  • 66 ÷ 33 = 2 (remainder 0)
  • HCF (231, 396) = 33

The challenge of finding the highest common factor (HCF) of 231 and 396 allows us to further refine our skills in applying prime factorization and the division method to larger numbers. Let's start by using the prime factorization method to break down 231 and 396 into their prime factors and identify common divisors. The prime factorization of 231 is 3 x 7 x 11, which indicates that 231 is composed of the prime numbers 3, 7, and 11. The prime factorization of 396 is 2 x 2 x 3 x 3 x 11, which can be written as 2² x 3² x 11. This shows that 396 is composed of the prime numbers 2, 3, and 11. To find the HCF, we identify the common prime factors with the lowest powers. Both numbers share the prime factors 3 and 11. The lowest power of 3 present in both factorizations is 3¹, and the lowest power of 11 is 11¹. Therefore, the HCF is the product of these common prime factors: 3 x 11 = 33. Now, let's use the division method, also known as the Euclidean algorithm, to verify our result. We begin by dividing the larger number (396) by the smaller number (231). 396 divided by 231 gives a quotient of 1 and a remainder of 165. Next, we divide 231 by the remainder 165, which gives a quotient of 1 and a remainder of 66. We continue this process by dividing 165 by the remainder 66. 165 divided by 66 gives a quotient of 2 and a remainder of 33. We then divide 66 by the remainder 33, which gives a quotient of 2 and a remainder of 0. Since the remainder is 0, the last non-zero divisor, which is 33, is the HCF. The HCF of 231 and 396, as determined by both the prime factorization method and the division method, is 33. This consistency underscores the reliability of these methods and demonstrates their effectiveness in solving HCF problems involving larger numbers. Working with numbers like 231 and 396 highlights the importance of these HCF-finding techniques in simplifying mathematical problems and gaining a deeper understanding of number theory.

Now, let's extend our understanding to finding the HCF of groups of three numbers.

(a) 28, 63 and 91

Prime Factorization Method

  • Prime factors of 28: 2 x 2 x 7 = 2² x 7
  • Prime factors of 63: 3 x 3 x 7 = 3² x 7
  • Prime factors of 91: 7 x 13
  • Common prime factors: 7
  • HCF (28, 63, 91) = 7

Division Method

  • First, find the HCF of 28 and 63:
    • 63 ÷ 28 = 2 (remainder 7)
    • 28 ÷ 7 = 4 (remainder 0)
    • HCF (28, 63) = 7
  • Now, find the HCF of 7 and 91:
    • 91 ÷ 7 = 13 (remainder 0)
    • HCF (28, 63, 91) = 7

The process of finding the highest common factor (HCF) of a group of three numbers, such as 28, 63, and 91, extends our understanding of HCF concepts and allows us to apply our skills in a more complex scenario. Let's begin by using the prime factorization method to break down each number into its prime factors. This will help us identify the common divisors among the three numbers. The prime factorization of 28 is 2 x 2 x 7, which can be written as 2² x 7. This indicates that 28 is composed of the prime numbers 2 and 7. The prime factorization of 63 is 3 x 3 x 7, or 3² x 7. This shows that 63 is composed of the prime numbers 3 and 7. The prime factorization of 91 is 7 x 13, which means that 91 is composed of the prime numbers 7 and 13. To find the HCF of the group, we look for the prime factors that are common to all three numbers. In this case, the only common prime factor is 7. Since 7 appears only once in the prime factorization of each number, the HCF is simply 7. Now, let's use the division method, also known as the Euclidean algorithm, to verify our result. When dealing with three numbers, we first find the HCF of any two numbers and then find the HCF of that result with the third number. First, we find the HCF of 28 and 63. Dividing 63 by 28 gives a quotient of 2 and a remainder of 7. Then, we divide 28 by the remainder 7, which gives a quotient of 4 and a remainder of 0. Thus, the HCF of 28 and 63 is 7. Next, we find the HCF of 7 (the HCF of 28 and 63) and 91. Dividing 91 by 7 gives a quotient of 13 and a remainder of 0. Therefore, the HCF of 7 and 91 is 7. This means that the HCF of the group 28, 63, and 91 is 7. The HCF of 28, 63, and 91, as determined by both the prime factorization method and the division method, is 7. This consistency reinforces the effectiveness of both methods and demonstrates their applicability to finding the HCF of groups of numbers. Working with three numbers showcases the utility of these HCF-finding techniques in simplifying mathematical problems and extending our understanding of number relationships.

(b) 60, 75 and 300

Prime Factorization Method

  • Prime factors of 60: 2 x 2 x 3 x 5 = 2² x 3 x 5
  • Prime factors of 75: 3 x 5 x 5 = 3 x 5²
  • Prime factors of 300: 2 x 2 x 3 x 5 x 5 = 2² x 3 x 5²
  • Common prime factors: 3 x 5
  • HCF (60, 75, 300) = 3 x 5 = 15

Division Method

  • First, find the HCF of 60 and 75:
    • 75 ÷ 60 = 1 (remainder 15)
    • 60 ÷ 15 = 4 (remainder 0)
    • HCF (60, 75) = 15
  • Now, find the HCF of 15 and 300:
    • 300 ÷ 15 = 20 (remainder 0)
    • HCF (60, 75, 300) = 15

The process of determining the highest common factor (HCF) of a group of three numbers, such as 60, 75, and 300, provides an excellent opportunity to reinforce our understanding of HCF concepts and apply our skills in a more complex context. Let's begin by employing the prime factorization method to break down each number into its prime factors. This approach will enable us to identify the common divisors shared among the three numbers. The prime factorization of 60 is 2 x 2 x 3 x 5, which can be expressed as 2² x 3 x 5. This indicates that 60 is composed of the prime numbers 2, 3, and 5. The prime factorization of 75 is 3 x 5 x 5, or 3 x 5². This reveals that 75 is composed of the prime numbers 3 and 5. The prime factorization of 300 is 2 x 2 x 3 x 5 x 5, which can be written as 2² x 3 x 5². This demonstrates that 300 is composed of the prime numbers 2, 3, and 5. To find the HCF of the group, we identify the prime factors that are common to all three numbers. In this case, the common prime factors are 3 and 5. Taking the lowest power of each common prime factor, we have 3¹ and 5¹. Therefore, the HCF is the product of these common prime factors: 3 x 5 = 15. Now, let's utilize the division method, also known as the Euclidean algorithm, to verify our result. When dealing with three numbers, we first find the HCF of any two numbers and then find the HCF of that result with the third number. First, we find the HCF of 60 and 75. Dividing 75 by 60 gives a quotient of 1 and a remainder of 15. Then, we divide 60 by the remainder 15, which gives a quotient of 4 and a remainder of 0. Thus, the HCF of 60 and 75 is 15. Next, we find the HCF of 15 (the HCF of 60 and 75) and 300. Dividing 300 by 15 gives a quotient of 20 and a remainder of 0. Therefore, the HCF of 15 and 300 is 15. This means that the HCF of the group 60, 75, and 300 is 15. The HCF of 60, 75, and 300, as determined by both the prime factorization method and the division method, is 15. This consistency reinforces the reliability of both methods and demonstrates their effectiveness in finding the HCF of groups of numbers. Working with three numbers showcases the utility of these HCF-finding techniques in simplifying mathematical problems and enhancing our understanding of number relationships.

(c) 48, 84 and 144

Prime Factorization Method

  • Prime factors of 48: 2 x 2 x 2 x 2 x 3 = 2⁴ x 3
  • Prime factors of 84: 2 x 2 x 3 x 7 = 2² x 3 x 7
  • Prime factors of 144: 2 x 2 x 2 x 2 x 3 x 3 = 2⁴ x 3²
  • Common prime factors: 2² x 3
  • HCF (48, 84, 144) = 2² x 3 = 12

Division Method

  • First, find the HCF of 48 and 84:
    • 84 ÷ 48 = 1 (remainder 36)
    • 48 ÷ 36 = 1 (remainder 12)
    • 36 ÷ 12 = 3 (remainder 0)
    • HCF (48, 84) = 12
  • Now, find the HCF of 12 and 144:
    • 144 ÷ 12 = 12 (remainder 0)
    • HCF (48, 84, 144) = 12

Determining the highest common factor (HCF) of a set of three numbers, such as 48, 84, and 144, provides us with an excellent opportunity to further apply our understanding of HCF principles and refine our problem-solving skills. Let's begin by employing the prime factorization method, which involves breaking down each number into its prime factors. This approach will enable us to identify the common divisors shared among the three numbers and determine their HCF. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, which can be expressed as 2⁴ x 3. This indicates that 48 is composed of the prime numbers 2 and 3. The prime factorization of 84 is 2 x 2 x 3 x 7, or 2² x 3 x 7. This reveals that 84 is composed of the prime numbers 2, 3, and 7. The prime factorization of 144 is 2 x 2 x 2 x 2 x 3 x 3, which can be written as 2⁴ x 3². This demonstrates that 144 is composed of the prime numbers 2 and 3. To find the HCF of the group, we identify the prime factors that are common to all three numbers. In this case, the common prime factors are 2 and 3. Taking the lowest power of each common prime factor, we have 2² and 3¹. Therefore, the HCF is the product of these common prime factors: 2² x 3 = 4 x 3 = 12. Now, let's utilize the division method, also known as the Euclidean algorithm, to verify our result. When dealing with three numbers, we first find the HCF of any two numbers and then find the HCF of that result with the third number. First, we find the HCF of 48 and 84. Dividing 84 by 48 gives a quotient of 1 and a remainder of 36. Then, we divide 48 by the remainder 36, which gives a quotient of 1 and a remainder of 12. Next, we divide 36 by the remainder 12, which gives a quotient of 3 and a remainder of 0. Thus, the HCF of 48 and 84 is 12. Next, we find the HCF of 12 (the HCF of 48 and 84) and 144. Dividing 144 by 12 gives a quotient of 12 and a remainder of 0. Therefore, the HCF of 12 and 144 is 12. This means that the HCF of the group 48, 84, and 144 is 12. The HCF of 48, 84, and 144, as determined by both the prime factorization method and the division method, is 12. This consistency reinforces the reliability of both methods and demonstrates their effectiveness in finding the HCF of groups of numbers. Working with three numbers showcases the utility of these HCF-finding techniques in simplifying mathematical problems and deepening our understanding of number relationships.

(d) 66, 154 and 187

Prime Factorization Method

  • Prime factors of 66: 2 x 3 x 11
  • Prime factors of 154: 2 x 7 x 11
  • Prime factors of 187: 11 x 17
  • Common prime factors: 11
  • HCF (66, 154, 187) = 11

Division Method

  • First, find the HCF of 66 and 154:
    • 154 ÷ 66 = 2 (remainder 22)
    • 66 ÷ 22 = 3 (remainder 0)
    • HCF (66, 154) = 22
  • Now, find the HCF of 22 and 187:
    • 187 ÷ 22 = 8 (remainder 11)
    • 22 ÷ 11 = 2 (remainder 0)
    • HCF (66, 154, 187) = 11

The task of finding the highest common factor (HCF) of the group of numbers 66, 154, and 187 allows us to apply our HCF skills to a more complex set of numbers and further solidify our understanding of different methodologies. Let's begin by employing the prime factorization method, which involves breaking down each number into its prime factors. This will enable us to identify the common divisors shared among the three numbers and determine their HCF. The prime factorization of 66 is 2 x 3 x 11, which indicates that 66 is composed of the prime numbers 2, 3, and 11. The prime factorization of 154 is 2 x 7 x 11, which reveals that 154 is composed of the prime numbers 2, 7, and 11. The prime factorization of 187 is 11 x 17, demonstrating that 187 is composed of the prime numbers 11 and 17. To find the HCF of the group, we identify the prime factors that are common to all three numbers. In this case, the only common prime factor is 11. Since 11 appears only once in the prime factorization of each number, the HCF is simply 11. Now, let's utilize the division method, also known as the Euclidean algorithm, to verify our result. When dealing with three numbers, we first find the HCF of any two numbers and then find the HCF of that result with the third number. First, we find the HCF of 66 and 154. Dividing 154 by 66 gives a quotient of 2 and a remainder of 22. Then, we divide 66 by the remainder 22, which gives a quotient of 3 and a remainder of 0. Thus, the HCF of 66 and 154 is 22. Next, we find the HCF of 22 (the HCF of 66 and 154) and 187. Dividing 187 by 22 gives a quotient of 8 and a remainder of 11. We then divide 22 by the remainder 11, which gives a quotient of 2 and a remainder of 0. Therefore, the HCF of 22 and 187 is 11. This means that the HCF of the group 66, 154, and 187 is 11. The HCF of 66, 154, and 187, as determined by both the prime factorization method and the division method, is 11. This consistency underscores the reliability of both methods and demonstrates their effectiveness in finding the HCF of groups of numbers. Working with these three numbers showcases the utility of these HCF-finding techniques in simplifying mathematical problems and deepening our understanding of number relationships.

Finding the highest common factor (HCF) is a fundamental skill in mathematics with numerous applications. Whether you use the prime factorization method or the division method, understanding the underlying principles is key to mastering this concept. By practicing with different pairs and groups of numbers, you can build confidence and proficiency in finding the HCF. The HCF, whether determined through prime factorization or the Euclidean algorithm, stands as a cornerstone of mathematical understanding, with its applications permeating various fields. As we've explored, the prime factorization method allows us to dissect numbers into their fundamental building blocks, revealing the common factors that contribute to their divisibility. This method provides a clear visual representation of the numbers' composition, enhancing our intuitive grasp of HCF. On the other hand, the Euclidean algorithm, with its iterative elegance, offers a systematic approach that efficiently narrows down possibilities until the HCF emerges. This method is particularly valuable when dealing with larger numbers, where prime factorization might become cumbersome. However, the true mastery of HCF lies not just in knowing the methods but in understanding the underlying principles. It's about recognizing the shared factors that bind numbers together, the common threads that weave through their divisibility. This understanding empowers us to not only calculate HCF but also to appreciate its significance in simplifying fractions, solving Diophantine equations, and tackling real-world problems. By practicing with diverse sets of numbers, we hone our skills, build confidence, and develop a deeper appreciation for the beauty and consistency of mathematical principles. The HCF, in essence, is more than just a number; it's a key to unlocking numerical relationships, a tool for simplifying complexity, and a testament to the power of mathematical thinking.