What Is The Gcf Of 84 And 90

What is the GCF of 84 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 concepts and different methods for solving it can be surprisingly insightful. This article delves deep into determining the GCF of 84 and 90, exploring various techniques and highlighting the importance of GCF in mathematics and beyond.

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 each of the integers without leaving a remainder. In simpler terms, it's the biggest number that goes into both numbers perfectly. For example, the GCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly.

Understanding the GCF is crucial in many mathematical operations, including simplifying fractions, solving algebraic equations, and working with geometric problems. Its practical applications extend beyond pure mathematics into fields like computer science and cryptography.

Methods for Finding the GCF of 84 and 90

Several methods can be employed to determine the GCF of 84 and 90. Let's explore three common approaches:

1. Listing Factors

This method involves listing all the factors of each number and identifying the largest common factor.

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

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

Comparing the lists, we can see that the common factors are 1, 2, 3, and 6. The greatest common factor is therefore 6.

This method is straightforward for smaller numbers but becomes cumbersome and time-consuming as the numbers get larger.

2. Prime Factorization

This method utilizes the prime factorization of each number. Prime factorization involves expressing a number as a product of its prime factors (numbers divisible only by 1 and themselves).

Prime factorization of 84: 2 x 2 x 3 x 7 = 2² x 3 x 7

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

To find the GCF, we identify the common prime factors and multiply them together, using the lowest power of each common factor. Both 84 and 90 share a '2' and a '3'. The lowest power of 2 is 2¹ (or simply 2), and the lowest power of 3 is 3¹. Therefore:

GCF(84, 90) = 2 x 3 = 6

This method is more efficient than listing factors, especially for larger numbers, as it provides a systematic approach to finding the GCF.

3. Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF of two numbers. It uses a series of divisions to iteratively reduce the problem until the GCF is found. The algorithm is 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.

Here's how the Euclidean algorithm works for 84 and 90:

1. Divide the larger number (90) by the smaller number (84): 90 ÷ 84 = 1 with a remainder of 6.
2. Replace the larger number with the remainder (6) and repeat the process using the previous smaller number (84) as the new larger number: 84 ÷ 6 = 14 with a remainder of 0.
3. Since the remainder is 0, the GCF is the last non-zero remainder, which is 6.

The Euclidean algorithm is particularly efficient for large numbers because it significantly reduces the number of calculations needed compared to the other methods. It's the preferred method for computer algorithms designed to find GCFs.

The Significance of GCF in Mathematics and Beyond

The GCF has broad applications across various mathematical fields and real-world scenarios. Some key areas include:

1. Simplifying Fractions

The GCF is essential for simplifying fractions to their lowest terms. To simplify a fraction, you divide both the numerator and the denominator by their GCF. For example, to simplify the fraction 84/90, we divide both the numerator and denominator by their GCF (6):

84 ÷ 6 = 14 90 ÷ 6 = 15

Therefore, 84/90 simplifies to 14/15.

2. Solving Algebraic Equations

GCF plays a role in factoring algebraic expressions. By factoring out the GCF from an expression, you can simplify it and solve equations more easily. For instance, in the expression 6x + 12, the GCF of 6x and 12 is 6. Factoring out the GCF gives us 6(x + 2).

3. Geometry and Measurement

GCF is used in geometric problems involving finding the dimensions of shapes. For example, to find the largest square tiles that can perfectly cover a rectangular floor with dimensions 84 cm and 90 cm, you need to find the GCF of 84 and 90, which is 6 cm. Therefore, the largest square tiles that can be used are 6 cm x 6 cm.

4. Computer Science and Cryptography

GCF algorithms are fundamental in computer science, especially in cryptography. The Euclidean algorithm, for example, is crucial in public-key cryptography, which forms the basis of secure online communication.

Conclusion: Mastering the GCF

Determining the greatest common factor of two numbers, like 84 and 90, is a fundamental skill in mathematics with far-reaching applications. While the simple method of listing factors might suffice for smaller numbers, the prime factorization and Euclidean algorithm provide more efficient and scalable methods for handling larger numbers. Understanding these methods empowers you to solve a wide range of mathematical problems and appreciate the GCF's significant role in various fields, from simplifying fractions to securing online transactions. The GCF, though seemingly simple, is a powerful concept that underpins many important mathematical and computational processes. Mastering its calculation is key to unlocking a deeper understanding of numerical relationships and their practical implications.