What Is The Gcf Of 75 And 90

What is the GCF of 75 and 90? A Deep Dive into Finding the Greatest Common Factor

Finding the greatest common factor (GCF) of two numbers might seem like a simple arithmetic task, but understanding the underlying principles and exploring different methods can reveal a surprisingly rich mathematical landscape. This article delves into the question: What is the GCF of 75 and 90? We'll not only answer this question but also explore multiple ways to find the GCF, discuss its applications, and delve into related mathematical concepts.

Understanding the Greatest Common Factor (GCF)

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides both of two or more numbers without leaving a remainder. It's a fundamental concept in number theory and has practical applications in various fields, from simplifying fractions to solving geometric problems. Understanding the GCF is crucial for grasping more advanced mathematical concepts.

For instance, if we consider the numbers 12 and 18, their factors are:

• 12: 1, 2, 3, 4, 6, 12
• 18: 1, 2, 3, 6, 9, 18

The common factors are 1, 2, 3, and 6. The greatest of these common factors is 6, therefore, the GCF of 12 and 18 is 6.

Methods for Finding the GCF of 75 and 90

Several methods can be used to determine the GCF of 75 and 90. Let's explore the most common ones:

1. Listing Factors

This is a straightforward method, especially for smaller numbers. We list all the factors of each number and identify the largest common factor.

Factors of 75: 1, 3, 5, 15, 25, 75 Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Comparing the two lists, we see that the common factors are 1, 3, 5, and 15. The greatest of these is 15. Therefore, the GCF of 75 and 90 is 15.

2. Prime Factorization

This method involves breaking down each number into its prime factors. The GCF is then found by multiplying the common prime factors raised to the lowest power.

Prime factorization of 75: 3 x 5 x 5 = 3 x 5² Prime factorization of 90: 2 x 3 x 3 x 5 = 2 x 3² x 5

The common prime factors are 3 and 5. The lowest power of 3 is 3¹ (or simply 3), and the lowest power of 5 is 5¹. Therefore, the GCF is 3 x 5 = 15.

3. Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF, particularly useful for larger numbers. It's based on the principle that the GCF of two numbers does not change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal.

Let's apply the Euclidean algorithm to 75 and 90:

1. 90 = 75 x 1 + 15
2. 75 = 15 x 5 + 0

Since the remainder is 0, the GCF is the last non-zero remainder, which is 15.

Applications of the Greatest Common Factor

The GCF has numerous applications across various mathematical domains and real-world scenarios:

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

• Solving Geometric Problems: The GCF is used in solving problems related to finding the dimensions of rectangular shapes that can be tiled perfectly with squares. For example, if you have a rectangular plot of land measuring 75 meters by 90 meters, you can determine the largest square tile that will perfectly cover the land without any leftover space (a tile of 15 meters by 15 meters).

• Number Theory: The GCF plays a vital role in various number theory concepts, such as modular arithmetic, Diophantine equations, and cryptography.

Beyond the GCF: Exploring Related Concepts

Understanding the GCF often leads to exploring related mathematical concepts:

• Least Common Multiple (LCM): The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers. The relationship between the GCF and LCM is given by the formula: GCF(a, b) x LCM(a, b) = a x b. For 75 and 90, the LCM is 450.

• Relative Primality: Two numbers are considered relatively prime (or coprime) if their GCF is 1. For example, 75 and 90 are not relatively prime because their GCF is 15.

• Modular Arithmetic: The GCF is essential in modular arithmetic, a system of arithmetic for integers where numbers "wrap around" upon reaching a certain value (the modulus).

Conclusion: Mastering the GCF and its Applications

This detailed exploration reveals that finding the GCF of 75 and 90, while seemingly simple, unlocks a deeper understanding of fundamental mathematical concepts. Whether you use the method of listing factors, prime factorization, or the Euclidean algorithm, the answer remains consistent: the GCF of 75 and 90 is 15. This seemingly simple calculation has far-reaching implications across various mathematical fields and real-world problems. By understanding the GCF and its related concepts, you expand your mathematical toolkit and gain valuable problem-solving skills applicable in numerous contexts. Remember to practice different methods to find the GCF to solidify your understanding and choose the method that best suits the numbers involved. This will empower you to tackle more complex mathematical challenges confidently.