Greatest Common Factor Of 75 And 90

Finding the Greatest Common Factor (GCF) of 75 and 90: A Comprehensive Guide

Finding the greatest common factor (GCF) of two numbers, like 75 and 90, might seem like a simple arithmetic task. However, understanding the underlying principles and various methods for calculating the GCF provides a strong foundation in number theory and is crucial for various mathematical applications. This comprehensive guide delves into the concept of GCF, explains multiple methods for calculating it, and explores its practical applications. We'll specifically focus on finding the GCF of 75 and 90, demonstrating each method step-by-step.

What is 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 each of the integers without leaving a remainder. In simpler terms, it's the biggest number that can be evenly divided 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. 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

There are several effective methods for determining the GCF, each with its own advantages and disadvantages. Let's explore the most common ones, applying them to find the GCF of 75 and 90:

1. Listing Factors Method

This method involves listing all the factors of each number and then identifying 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

Common Factors: 1, 3, 5, 15

Greatest Common Factor (GCF): 15

This method is straightforward for smaller numbers but becomes cumbersome and inefficient for larger numbers with many factors.

2. Prime Factorization Method

This method involves finding the prime factorization of each number and then multiplying the common prime factors raised to the lowest power.

Prime Factorization of 75:

75 = 3 x 25 = 3 x 5 x 5 = 3 x 5²

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

Common Prime Factors: 3 and 5

Lowest Powers: 3¹ and 5¹

GCF: 3¹ x 5¹ = 15

The prime factorization method is more efficient than the listing factors method, especially for larger numbers, as it systematically identifies the common factors.

3. Euclidean Algorithm

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

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

1. Start with the larger number (90) and the smaller number (75): 90, 75

2. Subtract the smaller number from the larger number: 90 - 75 = 15

3. Replace the larger number with the result (15): 75, 15

4. Repeat the process: 75 - 15 = 60

5. Replace the larger number: 60, 15

6. Repeat: 60 - 15 = 45

7. Replace: 45, 15

8. Repeat: 45 - 15 = 30

9. Replace: 30, 15

10. Repeat: 30 - 15 = 15

11. Replace: 15, 15

Since both numbers are now equal to 15, the GCF of 75 and 90 is 15.

A more efficient version of the Euclidean algorithm uses division instead of subtraction. We repeatedly divide the larger number by the smaller number and replace the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCF.

1. Divide 90 by 75: 90 = 1 x 75 + 15
2. Divide 75 by 15: 75 = 5 x 15 + 0

The last non-zero remainder is 15, so the GCF is 15. This version is generally preferred for its computational efficiency.

Applications of the Greatest Common Factor

The GCF has numerous applications in various fields, including:

• Simplifying Fractions: The GCF is used to simplify 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 Word Problems: Many word problems in mathematics involve finding the GCF to solve problems related to grouping, sharing, or dividing items evenly.

• Geometry: The GCF is used in geometry to find the dimensions of the largest square that can be used to tile a rectangle with given dimensions.

• Algebra: The GCF is crucial in factoring algebraic expressions, simplifying equations, and solving problems in abstract algebra.

• Computer Science: The GCF is used in cryptography, computer graphics, and other areas of computer science.

• Music Theory: Understanding GCF helps in simplifying musical rhythms and finding common time signatures.

Conclusion: The GCF of 75 and 90 is 15

Through the various methods explored – listing factors, prime factorization, and the Euclidean algorithm – we have conclusively determined that the greatest common factor of 75 and 90 is 15. Understanding these methods empowers you to efficiently calculate the GCF of any two integers, providing a valuable tool for various mathematical applications. Remember to choose the method most suitable for the numbers involved, prioritizing efficiency and accuracy. The Euclidean algorithm, particularly its division-based version, offers the most efficient approach for larger numbers. Mastering the GCF calculation not only strengthens your arithmetic skills but also unlocks a deeper understanding of fundamental mathematical concepts.