Greatest Common Factor Of 72 And 90

Finding the Greatest Common Factor (GCF) of 72 and 90: A Comprehensive Guide

Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics with applications spanning various fields, from simplifying fractions to solving algebraic equations. This comprehensive guide will delve into multiple methods for determining the GCF of 72 and 90, explaining each approach thoroughly and highlighting their strengths and weaknesses. We'll also explore the broader significance of GCFs and their practical uses.

Understanding the Greatest Common Factor (GCF)

Before we tackle the specific case of 72 and 90, let's solidify our understanding of the GCF. The GCF of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. In simpler terms, it's the biggest number that goes evenly into both numbers.

For example, 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. Therefore, the greatest common factor of 12 and 18 is 6.

Method 1: Listing Factors

This is the most straightforward method, particularly suitable for smaller numbers. We list all the factors of each number and then identify the largest factor common to both.

Factors of 72:

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

Factors of 90:

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

Comparing the two lists, we find the common factors are 1, 2, 3, 6, 9, and 18. The greatest of these common factors is 18. Therefore, the GCF of 72 and 90 is 18.

Method 2: Prime Factorization

This method involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves. It's a more efficient approach for larger numbers.

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

Prime Factorization of 90:

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

Now, we identify the common prime factors and their lowest powers:

• Both numbers have a factor of 2 (to the power of 1, since 2¹ is the lowest power present in both factorizations).
• Both numbers have a factor of 3 (to the power of 2, since 3² is the lowest power present in both factorizations).

To find the GCF, we multiply these common prime factors with their lowest powers:

GCF(72, 90) = 2¹ x 3² = 2 x 9 = 18

Method 3: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF of two numbers, especially useful 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, and that number is the GCF.

1. Start with the larger number (90) and the smaller number (72).
2. Divide the larger number by the smaller number and find the remainder: 90 ÷ 72 = 1 with a remainder of 18.
3. Replace the larger number with the smaller number (72) and the smaller number with the remainder (18).
4. Repeat step 2: 72 ÷ 18 = 4 with a remainder of 0.
5. Since the remainder is 0, the GCF is the last non-zero remainder, which is 18.

Therefore, the GCF(72, 90) = 18.

Applications of the Greatest Common Factor

The GCF has numerous applications in various areas of mathematics and beyond:

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

• Solving Algebraic Equations: GCF plays a role in factoring polynomials, a fundamental skill in algebra. Finding the GCF of the terms in a polynomial allows you to factor it, making it easier to solve equations.

• Measurement and Geometry: GCF helps in solving problems related to measurement and geometry. For example, finding the largest square tile that can perfectly cover a rectangular floor with dimensions 72 units and 90 units requires finding the GCF of 72 and 90. The answer, 18, represents the side length of the largest square tile.

• Number Theory: GCF is a core concept in number theory, the branch of mathematics concerned with the properties of integers. It's used in various theorems and proofs related to prime numbers, divisibility, and modular arithmetic.

• Computer Science: The Euclidean algorithm for finding the GCF is used in computer science algorithms for various tasks, including cryptography and data compression.

Beyond the Basics: Extending the Concept

The GCF isn't limited to just two numbers. You can find the GCF of multiple numbers using the same principles. One effective method is to find the prime factorization of each number and then identify the common prime factors and their lowest powers. The product of these common prime factors with their lowest powers is the GCF of all the numbers.

For instance, to find the GCF of 72, 90, and 108:

• Prime factorization of 72: 2³ x 3²
• Prime factorization of 90: 2 x 3² x 5
• Prime factorization of 108: 2² x 3³

The common prime factors are 2 and 3. The lowest power of 2 is 2¹ and the lowest power of 3 is 3². Therefore, the GCF(72, 90, 108) = 2¹ x 3² = 18.

Conclusion

Finding the greatest common factor is a seemingly simple yet profoundly important mathematical skill. Mastering various methods, from listing factors to utilizing the efficient Euclidean algorithm, empowers you to solve a wide range of problems across different mathematical disciplines and real-world applications. Understanding the GCF is essential for anyone seeking a deeper understanding of numbers and their relationships. This knowledge forms a solid foundation for more advanced mathematical concepts and problem-solving strategies. Whether simplifying fractions, factoring polynomials, or tackling geometrical challenges, the GCF remains a fundamental tool in your mathematical toolbox. Practice these methods regularly to strengthen your understanding and improve your proficiency.