Highest Common Factor Of 48 And 42

Finding the Highest Common Factor (HCF) of 48 and 42: A Comprehensive Guide

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides exactly into two or more numbers without leaving a remainder. Finding the HCF is a fundamental concept in mathematics with applications in various fields, from simplifying fractions to solving complex algebraic problems. This article delves deep into finding the HCF of 48 and 42, exploring multiple methods and demonstrating their practical application. We'll also touch upon the broader significance of HCF in mathematics and beyond.

Understanding the Concept of HCF

Before we dive into the calculation, let's solidify our understanding of the HCF. The HCF represents the greatest common divisor shared by two or more numbers. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors shared by both 12 and 18 are 1, 2, 3, and 6. Therefore, the HCF of 12 and 18 is 6.

Key characteristics of the HCF:

• Divisibility: The HCF must divide both numbers without leaving a remainder.
• Largest Common Factor: It's the largest number that satisfies the divisibility condition.
• Uniqueness: For any given set of numbers, there's only one HCF.

Now, let's focus on finding the HCF of 48 and 42.

Method 1: Prime Factorization Method

This method involves breaking down each number into its prime factors and then identifying the common prime factors raised to the lowest power. Prime factors are numbers divisible only by 1 and themselves (e.g., 2, 3, 5, 7, 11...).

Steps:

1. Find the prime factorization of 48:

   48 = 2 x 24 = 2 x 2 x 12 = 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 3 = 2⁴ x 3¹

2. Find the prime factorization of 42:

   42 = 2 x 21 = 2 x 3 x 7 = 2¹ x 3¹ x 7¹

3. Identify common prime factors: Both 48 and 42 share the prime factors 2 and 3.

4. Determine the lowest power: The lowest power of 2 is 2¹ (from 42's factorization), and the lowest power of 3 is 3¹ (from both factorizations).

5. Calculate the HCF: Multiply the common prime factors raised to their lowest powers: 2¹ x 3¹ = 2 x 3 = 6

Therefore, the HCF of 48 and 42 is 6.

Method 2: Euclidean Algorithm

The Euclidean algorithm is an efficient method for finding the HCF of two numbers, especially when dealing with larger numbers. It's based on the principle that the HCF of two numbers doesn't change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers become equal, and that number is the HCF.

Steps:

1. Start with the larger number (48) and the smaller number (42):

   48, 42

2. Subtract the smaller number from the larger number:

   48 - 42 = 6

3. Replace the larger number with the result (6), and keep the smaller number (42):

   6, 42

4. Repeat the process:

   42 - 6 x 7 = 0 (We can see that 6 goes into 42 seven times exactly)

5. The last non-zero remainder is the HCF: The last non-zero remainder is 6.

Therefore, the HCF of 48 and 42 is 6.

Method 3: Listing Factors Method

This method is suitable for smaller numbers and involves listing all the factors of each number and identifying the largest common factor.

Steps:

1. List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

2. List the factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

3. Identify common factors: The common factors are 1, 2, 3, and 6.

4. The largest common factor is the HCF: The largest common factor is 6.

Therefore, the HCF of 48 and 42 is 6.

Comparing the Methods

All three methods yield the same result: the HCF of 48 and 42 is 6. However, each method has its advantages and disadvantages:

• Prime Factorization: Effective for smaller numbers, but can become cumbersome for very large numbers as finding prime factors can be time-consuming. It provides a good understanding of the number's composition.

• Euclidean Algorithm: Highly efficient for any size of numbers, even very large ones. It's a computationally faster method.

• Listing Factors: Suitable only for smaller numbers; it becomes impractical for larger numbers because listing all factors can be very lengthy.

Applications of HCF

The HCF finds practical applications in various areas:

• Simplifying Fractions: Finding the HCF of the numerator and denominator helps in simplifying fractions to their lowest terms. For example, the fraction 48/42 can be simplified to 8/7 by dividing both numerator and denominator by their HCF, which is 6.

• Solving Word Problems: Many word problems involving sharing, grouping, or dividing quantities require finding the HCF to determine the largest possible equal groups or shares.

• Algebra and Number Theory: HCF plays a crucial role in various algebraic and number theory concepts, such as solving Diophantine equations and finding least common multiples (LCM). The LCM and HCF are closely related; their product equals the product of the two original numbers.

• Geometry and Measurement: HCF helps in solving problems related to finding the greatest possible side length of identical squares that can be formed from a rectangular area.

Conclusion

Finding the HCF of two numbers is a fundamental skill in mathematics with broad applications. We've explored three different methods – prime factorization, the Euclidean algorithm, and listing factors – each with its strengths and weaknesses. The Euclidean algorithm stands out as the most efficient method for larger numbers. Understanding the concept of HCF and mastering the different calculation methods are essential for anyone pursuing further studies in mathematics or related fields. The ability to find the HCF efficiently is a key building block for more advanced mathematical concepts and problem-solving. The example of finding the HCF of 48 and 42 serves as a practical illustration of these fundamental mathematical principles and their real-world relevance. Remember to choose the method that best suits your needs and the size of the numbers involved.