What Is The Greatest Common Factor Of 75 And 90

What is the Greatest Common Factor of 75 and 90? A Deep Dive into Number Theory

Finding the greatest common factor (GCF) of two numbers might seem like a simple arithmetic problem, but understanding the underlying concepts unlocks a world of mathematical elegance and practical applications. This article will explore the GCF of 75 and 90, examining multiple methods for calculating it and delving into the theoretical underpinnings that make this seemingly basic operation so significant.

Understanding Greatest Common Factors (GCF)

The greatest common factor, also known as the greatest common divisor (GCD), is the largest number that divides evenly into two or more numbers without leaving a remainder. It's a fundamental concept in number theory, impacting areas like simplifying fractions, solving algebraic equations, and even cryptography.

Think of it like finding the largest shared building block of two numbers. Just as you can break down a building into its bricks, you can break down numbers into their prime factors. The GCF is the collection of those shared prime factors raised to the lowest power they appear in either number.

Method 1: Prime Factorization

This method is considered the most fundamental and conceptually clearest way to find the GCF. It involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves.

Step 1: Find the Prime Factorization of 75

75 can be factored as:

• 75 = 3 x 25
• 75 = 3 x 5 x 5
• 75 = 3 x 5²

Step 2: Find the Prime Factorization of 90

90 can be factored as:

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

Step 3: Identify Common Prime Factors

Comparing the prime factorizations of 75 (3 x 5²) and 90 (2 x 3² x 5), we see that they share the prime factors 3 and 5.

Step 4: Determine the Lowest Power

The lowest power of 3 that appears in both factorizations is 3¹ (or simply 3). The lowest power of 5 that appears is 5¹.

Step 5: Calculate the GCF

Multiply the common prime factors raised to their lowest powers: 3¹ x 5¹ = 15

Therefore, the greatest common factor of 75 and 90 is 15.

Method 2: Euclidean Algorithm

The Euclidean algorithm provides a more efficient method, especially for larger numbers. This iterative method relies on repeated division until the remainder is 0. The last non-zero remainder is the GCF.

Step 1: Divide the larger number by the smaller number

90 ÷ 75 = 1 with a remainder of 15

Step 2: Replace the larger number with the smaller number, and the smaller number with the remainder

Now we work with 75 and 15.

75 ÷ 15 = 5 with a remainder of 0

Step 3: The GCF is the last non-zero remainder

Since the remainder is 0, the GCF is the previous remainder, which is 15.

The Euclidean algorithm is remarkably efficient because it avoids the need for complete prime factorization, making it a preferred method for large numbers where prime factorization can become computationally intensive.

Method 3: Listing Factors

This method is best suited for smaller numbers and provides a good intuitive understanding of the concept.

Step 1: List all factors of 75

1, 3, 5, 15, 25, 75

Step 2: List all factors of 90

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

Step 3: Identify common factors

The common factors of 75 and 90 are 1, 3, 5, and 15.

Step 4: Determine the greatest common factor

The largest common factor is 15.

While straightforward, this method becomes impractical for larger numbers due to the increasing number of factors to consider.

Applications of the GCF

The seemingly simple task of finding the GCF has far-reaching applications in various mathematical fields and practical scenarios:

• Simplifying Fractions: The GCF is crucial for reducing fractions to their simplest form. For example, the fraction 75/90 can be simplified to 5/6 by dividing both the numerator and denominator by their GCF (15).

• Solving Linear Diophantine Equations: These equations involve finding integer solutions to equations of the form ax + by = c. The GCF of 'a' and 'b' plays a vital role in determining the existence and nature of solutions.

• Modular Arithmetic and Cryptography: The GCF is fundamental in understanding modular arithmetic, which is the basis of many cryptographic systems used to secure online communications.

• Least Common Multiple (LCM): The GCF is closely related to the least common multiple (LCM). The product of the GCF and LCM of two numbers is always equal to the product of the two numbers. This relationship is invaluable in solving various mathematical problems involving fractions and ratios.

Beyond 75 and 90: Extending the Concepts

The techniques discussed above – prime factorization, the Euclidean algorithm, and listing factors – are applicable to finding the GCF of any two (or more) numbers. The choice of method depends on the size of the numbers involved and the desired level of understanding. For smaller numbers, listing factors offers a clear visual approach, while the Euclidean algorithm's efficiency shines for larger numbers. Understanding prime factorization provides a fundamental theoretical grounding.

Furthermore, the concepts extend to finding the GCF of more than two numbers. You can apply the chosen method iteratively, finding the GCF of two numbers, then finding the GCF of the result and the next number, and so on. For example, to find the GCF of 75, 90, and 105, you would first find the GCF of 75 and 90 (which is 15), and then find the GCF of 15 and 105. This process would yield a final GCF of 15.

Conclusion: The Significance of the GCF

The seemingly simple question, "What is the greatest common factor of 75 and 90?", opens a door to a rich tapestry of mathematical concepts and applications. Mastering the calculation of the GCF, along with understanding its underlying principles, is essential for anyone pursuing a deeper understanding of mathematics and its practical implications. Whether you're simplifying fractions, exploring number theory, or delving into the intricacies of cryptography, the GCF is an indispensable tool. This article has explored different methods to calculate the GCF and highlighted its broader significance in mathematics and beyond, demonstrating that even the most basic mathematical operations can hold remarkable depth and utility.