Highest Common Factor Of 84 And 24

Finding the Highest Common Factor (HCF) of 84 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 across various fields, from simplifying fractions to solving complex mathematical problems. This article will explore several methods to determine the HCF of 84 and 24, explaining each approach in detail and highlighting their practical applications. We'll also delve into the broader context of HCF, its importance, and related concepts.

Understanding the Concept of Highest Common Factor (HCF)

Before we dive into calculating the HCF of 84 and 24, let's solidify our understanding of the concept. The HCF is essentially the largest common factor shared by two or more numbers. A factor is a number that divides another number completely without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

To find the HCF, we need to identify all the factors of each number and then select the largest factor common to both. This method works well for smaller numbers but becomes cumbersome for larger ones. Therefore, we'll explore more efficient methods below.

Method 1: Prime Factorization Method

The prime factorization method involves expressing each number as a product of its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Once we have the prime factorization of both numbers, the HCF is the product of the common prime factors raised to the lowest power.

Let's apply this method to find the HCF of 84 and 24:

1. Prime Factorization of 84:

84 = 2 x 42 = 2 x 2 x 21 = 2 x 2 x 3 x 7 = 2² x 3 x 7

2. Prime Factorization of 24:

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

3. Identifying Common Prime Factors:

Both 84 and 24 have the prime factors 2 and 3.

4. Determining the HCF:

The lowest power of 2 is 2¹, and the lowest power of 3 is 3¹. Therefore, the HCF of 84 and 24 is 2¹ x 3¹ = 6.

Method 2: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the HCF of two 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 we reach a point where the remainder is zero. The last non-zero remainder is the HCF.

Let's use the Euclidean algorithm to find the HCF of 84 and 24:

1. Divide the larger number (84) by the smaller number (24):

84 ÷ 24 = 3 with a remainder of 12

2. Replace the larger number with the remainder:

Now we find the HCF of 24 and 12.

3. Repeat the process:

24 ÷ 12 = 2 with a remainder of 0

4. The HCF is the last non-zero remainder:

Since the remainder is 0, the HCF is the previous remainder, which is 12. There's a mistake here in the application; let's correct it.

Let's redo the Euclidean Algorithm correctly:

84 ÷ 24 = 3 remainder 12
24 ÷ 12 = 2 remainder 0

Therefore, the HCF of 84 and 24 is 12. My apologies for the previous error.

Method 3: Listing Factors Method

This method, while less efficient for larger numbers, is conceptually straightforward. We list all the factors of each number and then identify the largest common factor.

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Common Factors: 1, 2, 3, 4, 6, 12

Highest Common Factor: 12

Applications of HCF

The HCF finds applications in various areas, including:

Simplifying Fractions: The HCF is used to simplify fractions to their lowest terms. For example, the fraction 84/24 can be simplified by dividing both the numerator and denominator by their HCF (12), resulting in the simplified fraction 7/2.
Solving Word Problems: Many word problems involving division and common factors require finding the HCF. For instance, problems related to distributing items equally among groups or finding the largest possible size of identical square tiles that can cover a rectangular area.
Number Theory: The HCF is a fundamental concept in number theory, used in various advanced mathematical theorems and proofs.
Computer Science: The Euclidean algorithm, used to calculate the HCF, is a core algorithm in computer science, optimized for its efficiency in handling large numbers.
Cryptography: HCF plays a role in certain cryptographic algorithms, particularly those involving modular arithmetic.

Understanding Least Common Multiple (LCM) in Relation to HCF

The least common multiple (LCM) is another important concept closely related to HCF. The LCM of two numbers is the smallest number that is a multiple of both numbers. There's a useful relationship between the HCF and LCM of two numbers:

HCF(a, b) x LCM(a, b) = a x b

This formula allows us to calculate the LCM of two numbers if we know their HCF, or vice-versa. For 84 and 24, where HCF is 12:

LCM(84, 24) = (84 x 24) / 12 = 168

Conclusion

Finding the highest common factor is a crucial skill in mathematics with widespread applications. We've explored three different methods to calculate the HCF of 84 and 24 – prime factorization, the Euclidean algorithm, and listing factors. The Euclidean algorithm is generally the most efficient for larger numbers. Understanding HCF lays the foundation for grasping more advanced concepts in number theory and its applications in various fields. Remember, practice is key to mastering these techniques and developing a strong intuition for number relationships. By understanding the underlying principles and applying the correct methods, you can confidently tackle HCF problems of any complexity.