Highest Common Factor Of 72 And 108

Finding the Highest Common Factor (HCF) of 72 and 108: A Comprehensive Guide

Finding the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two numbers is a fundamental concept in mathematics with applications ranging from simplifying fractions to solving complex algebraic problems. This article will delve deep into determining the HCF of 72 and 108, exploring various methods and illustrating the underlying principles. We'll go beyond a simple answer and explore the broader mathematical concepts involved.

Understanding Highest Common Factor (HCF)

The HCF of two or more numbers is the largest number that divides each of them without leaving a remainder. It's the biggest common divisor shared by all the 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 of 12 and 18 are 1, 2, 3, and 6. The highest of these common factors is 6; therefore, the HCF of 12 and 18 is 6.

Methods for Finding the HCF of 72 and 108

Several methods exist for calculating the HCF. Let's explore the most common approaches, applying them to find the HCF of 72 and 108:

1. Prime Factorization Method

This method involves expressing each number as a product of its prime factors. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11).

Steps:

1. Find the prime factorization of 72: 72 = 2 x 36 = 2 x 2 x 18 = 2 x 2 x 2 x 9 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

2. Find the prime factorization of 108: 108 = 2 x 54 = 2 x 2 x 27 = 2 x 2 x 3 x 9 = 2 x 2 x 3 x 3 x 3 = 2² x 3³

3. Identify common prime factors: Both 72 and 108 have 2² and 3² as common prime factors.

4. Multiply the common prime factors: 2² x 3² = 4 x 9 = 36

Therefore, the HCF of 72 and 108 is 36.

2. Division Method (Euclidean Algorithm)

The Euclidean algorithm is an efficient method for finding the HCF of two numbers. It repeatedly applies the division algorithm until the remainder is zero. The last non-zero remainder is the HCF.

Steps:

1. Divide the larger number (108) by the smaller number (72): 108 ÷ 72 = 1 with a remainder of 36

2. Replace the larger number with the smaller number (72) and the smaller number with the remainder (36): 72 ÷ 36 = 2 with a remainder of 0

Since the remainder is 0, the last non-zero remainder (36) is the HCF.

Therefore, the HCF of 72 and 108 is 36.

3. Listing Factors Method

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.

Steps:

1. List the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

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

3. Identify the common factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

4. The largest common factor is 36.

Therefore, the HCF of 72 and 108 is 36.

Applications of HCF

Understanding and calculating the HCF has numerous applications in various mathematical fields and real-world scenarios:

• Simplifying Fractions: The HCF is crucial for simplifying fractions to their lowest terms. For example, the fraction 72/108 can be simplified by dividing both the numerator and denominator by their HCF (36), resulting in the simplified fraction 2/3.

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

• Geometry: The HCF is used in geometry to find the dimensions of the largest square tile that can perfectly cover a rectangular area.

• Number Theory: The HCF plays a significant role in various number theory concepts, including modular arithmetic and Diophantine equations.

Further Exploration: Least Common Multiple (LCM)

While this article focuses on HCF, it's important to understand its relationship with the Least Common Multiple (LCM). The LCM of two numbers is the smallest number that is a multiple of both numbers. There's a vital relationship between the HCF and LCM:

For any two numbers 'a' and 'b', HCF(a, b) x LCM(a, b) = a x b

Using this relationship, we can calculate the LCM of 72 and 108:

HCF(72, 108) x LCM(72, 108) = 72 x 108

36 x LCM(72, 108) = 7776

LCM(72, 108) = 7776 / 36 = 216

Therefore, the LCM of 72 and 108 is 216.

Conclusion

Finding the HCF of 72 and 108, whether through prime factorization, the Euclidean algorithm, or listing factors, consistently yields the result of 36. This simple calculation highlights a fundamental concept with far-reaching implications in mathematics and problem-solving. Understanding the different methods and the relationship between HCF and LCM empowers you to tackle various mathematical challenges effectively. Mastering these concepts forms a solid foundation for more advanced mathematical studies. Remember to practice these methods with different numbers to solidify your understanding and improve your problem-solving skills. The more you practice, the quicker and more efficient you will become at finding the HCF of any two numbers.