Highest Common Factor Of 108 And 24

Finding the Highest Common Factor (HCF) of 108 and 24: 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 number theory with applications in various fields, including cryptography and computer science. This article will explore multiple methods to determine the HCF of 108 and 24, offering a deep dive into the process and providing a solid understanding of the underlying principles.

Understanding the Concept of HCF

Before we delve into calculating the HCF of 108 and 24, let's solidify our understanding of what the HCF represents. Imagine you have 108 apples and 24 oranges. You want to divide both fruits into identical groups, with each group containing the same number of apples and oranges, and without any fruits left over. The HCF represents the maximum number of such identical groups you can create. In this case, finding the HCF of 108 and 24 will tell us the largest possible number of groups we can make.

Method 1: Prime Factorization

The prime factorization method is a robust and widely used technique for finding the HCF. It involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves.

Step 1: Prime Factorization of 108

Let's start by finding the prime factors of 108:

108 = 2 x 54 = 2 x 2 x 27 = 2 x 2 x 3 x 9 = 2² x 3³

Therefore, the prime factorization of 108 is 2² x 3³.

Step 2: Prime Factorization of 24

Next, we find the prime factors of 24:

24 = 2 x 12 = 2 x 2 x 6 = 2 x 2 x 2 x 3 = 2³ x 3

Therefore, the prime factorization of 24 is 2³ x 3.

Step 3: Identifying Common Factors

Now, we identify the common prime factors in both factorizations:

• Both 108 and 24 contain the prime factor 2.
• Both 108 and 24 contain the prime factor 3.

Step 4: Calculating the HCF

The HCF is the product of the lowest powers of the common prime factors. In this case:

• The lowest power of 2 is 2¹ (or simply 2).
• The lowest power of 3 is 3¹.

Therefore, the HCF of 108 and 24 is 2 x 3 = 6.

Method 2: Listing Factors

This method involves listing all the factors of each number and then identifying the largest common factor. While straightforward for smaller numbers, it becomes less efficient for larger numbers.

Step 1: Factors of 108

The factors of 108 are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.

Step 2: Factors of 24

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.

Step 3: Identifying Common Factors

Comparing the two lists, the common factors are: 1, 2, 3, 4, 6, 12.

Step 4: Determining the HCF

The largest common factor is 12. Note: There seems to be a discrepancy here. The prime factorization method yielded an HCF of 6, while the listing method yielded 12. The prime factorization method is generally more reliable, especially with larger numbers, and we should double-check our work. There is an error in the listing of factors for 108. The correct list should include 6 but not 12. The largest common factor is indeed 6.

Method 3: Euclidean Algorithm

The Euclidean algorithm is an efficient method for finding the HCF of two numbers, particularly useful for larger numbers where prime factorization becomes cumbersome. 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 are equal, and that number is the HCF.

Step 1: Initial Setup

We start with the two numbers: 108 and 24.

Step 2: Repeated Subtraction (or Division with Remainder)

We can either repeatedly subtract the smaller number from the larger number or use division with remainder:

• Division with Remainder: 108 ÷ 24 = 4 with a remainder of 12
• Now, we replace the larger number (108) with the remainder (12) and repeat the process: 24 ÷ 12 = 2 with a remainder of 0

Step 3: Determining the HCF

When the remainder is 0, the HCF is the last non-zero remainder. In this case, the last non-zero remainder is 12. Again, we have a discrepancy. Let's review the steps. The division should be:

• 108 ÷ 24 = 4 remainder 12
• 24 ÷ 12 = 2 remainder 0

The last non-zero remainder is 12. There appears to be a consistent error in the calculations above. Let's re-examine the factor list for 108. The correct factors are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108. The common factors between 108 and 24 are 1, 2, 3, 4, 6, 12. The greatest common factor is indeed 12. Our previous calculations using prime factorization and the initial Euclidean algorithm had errors. The correct HCF is 12.

Applications of HCF

The HCF has practical applications in various areas:

• Simplifying Fractions: The HCF helps in simplifying fractions to their lowest terms. For example, the fraction 108/24 can be simplified by dividing both numerator and denominator by their HCF (12), resulting in the simplified fraction 9/2.

• Solving Word Problems: Many word problems involving grouping or division utilize the concept of HCF. For example, problems related to dividing items into equal groups, distributing resources evenly, or determining the largest possible size of identical squares that can be formed from a rectangular area often require finding the HCF.

• Cryptography: The HCF plays a crucial role in public-key cryptography algorithms. The security of these algorithms depends heavily on the difficulty of finding the HCF of very large numbers.

• Computer Science: HCF calculations are fundamental in computer algorithms related to number theory, data structures, and optimization.

Conclusion

Finding the highest common factor (HCF) of two numbers is a fundamental concept in mathematics with practical applications across multiple disciplines. While several methods exist to determine the HCF, the prime factorization method provides a clear and methodical approach, especially for understanding the underlying principle. The Euclidean algorithm is highly efficient for larger numbers, offering a quicker route to the solution. Regardless of the method chosen, understanding the concept of HCF is crucial for solving numerous problems involving number theory and its practical applications. We have explored multiple methods, corrected errors in our initial calculations, and arrived at the accurate conclusion: the HCF of 108 and 24 is 12.