What Is The Greatest Common Factor Of 72 And 90

What is the Greatest Common Factor of 72 and 90? A Deep Dive into Number Theory

Finding the greatest common factor (GCF) of two numbers might seem like a simple arithmetic task, but it's a concept that underpins many areas of mathematics, from simplifying fractions to solving algebraic equations. This article will explore the various methods for determining the GCF of 72 and 90, delve into the underlying mathematical principles, and demonstrate its practical applications. We’ll go beyond simply providing the answer; we’ll provide a thorough understanding of the process and its relevance in a broader mathematical context.

Understanding Greatest Common Factor (GCF)

The greatest common factor (GCF), also known as the greatest common divisor (GCD), of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. It’s essentially the largest number that is a factor of both numbers. This concept is fundamental in simplifying fractions, solving equations, and understanding the relationships between numbers.

Method 1: Prime Factorization

This method is considered a classic approach for finding the GCF. It involves breaking down each number into its prime factors. Prime factors are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...).

Step 1: Find the prime factorization of 72.

72 can be factored as follows:

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

Step 2: Find the prime factorization of 90.

90 can be factored as follows:

90 = 2 x 45 = 2 x 3 x 15 = 2 x 3 x 3 x 5 = 2 x 3² x 5

Step 3: Identify common prime factors.

Both 72 and 90 share the prime factors 2 and 3.

Step 4: Determine the GCF.

The lowest power of the common prime factors is 2¹ and 3². Therefore, the GCF is 2¹ x 3² = 2 x 9 = 18.

Therefore, the greatest common factor of 72 and 90 is 18.

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: List the factors of 72.

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Step 2: List the factors of 90.

Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Step 3: Identify common factors.

Common factors of 72 and 90: 1, 2, 3, 6, 9, 18

Step 4: Determine the GCF.

The largest common factor is 18.

Therefore, the greatest common factor of 72 and 90 is 18.

Method 3: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF, especially for larger numbers. It’s based on the principle that the GCF 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.

Step 1: Divide the larger number (90) by the smaller number (72) and find the remainder.

90 ÷ 72 = 1 with a remainder of 18

Step 2: Replace the larger number with the remainder (18).

Now we find the GCF of 72 and 18.

Step 3: Repeat the process.

72 ÷ 18 = 4 with a remainder of 0

Since the remainder is 0, the GCF is the last non-zero remainder, which is 18.

Therefore, the greatest common factor of 72 and 90 is 18.

Applications of GCF

The concept of the greatest common factor has numerous applications across various mathematical fields and real-world scenarios:

• Simplifying Fractions: The GCF is crucial for reducing fractions to their simplest form. For instance, the fraction 72/90 can be simplified by dividing both the numerator and denominator by their GCF (18), resulting in the equivalent fraction 4/5.

• Solving Equations: GCF plays a role in solving Diophantine equations, which are equations where only integer solutions are sought.

• Geometry: GCF is used in geometry problems related to finding the largest square that can tile a rectangle with given dimensions.

• Number Theory: GCF is a fundamental concept in number theory, which explores the properties of integers. It's involved in concepts like modular arithmetic and prime factorization.

• Real-World Applications: GCF can be applied in various real-world situations, such as dividing items evenly among groups, determining the size of the largest square tile that can fit perfectly into a rectangular area, or finding the common time interval for recurring events.

Beyond the Basics: Exploring Related Concepts

Understanding the GCF opens doors to other important mathematical concepts:

• Least Common Multiple (LCM): The LCM is the smallest positive integer that is a multiple of both numbers. There's a relationship between the GCF and LCM: The product of the GCF and LCM of two numbers is equal to the product of the two numbers. For 72 and 90, the LCM is 360 (72 x 90 = 18 x 360).

• Coprime Numbers: Two numbers are considered coprime (or relatively prime) if their GCF is 1. For example, 7 and 15 are coprime.

Conclusion

Finding the greatest common factor of 72 and 90, as demonstrated through various methods, highlights the significance of this fundamental concept in mathematics. The prime factorization method provides a deep understanding of the numbers' structure, while the Euclidean algorithm offers efficiency for larger numbers. Understanding the GCF not only facilitates solving mathematical problems but also extends to various practical applications across numerous disciplines. The ability to efficiently find the GCF is a valuable skill for anyone pursuing studies or working in fields that involve mathematical reasoning and problem-solving. From simplifying fractions to solving complex equations, the GCF remains a cornerstone of mathematical understanding. The connection between GCF and LCM further deepens our understanding of the relationships between numbers and provides valuable tools for solving a wide range of mathematical problems.