Greatest Common Factor Of 84 And 90

Finding the Greatest Common Factor (GCF) of 84 and 90: A Comprehensive Guide

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. Finding the GCF is a fundamental concept in mathematics, crucial for simplifying fractions, solving algebraic equations, and understanding number theory. This article will explore multiple methods for determining the GCF of 84 and 90, explaining the underlying principles and providing practical examples. We'll delve into the prime factorization method, the Euclidean algorithm, and the listing factors method, offering a comprehensive understanding of this important mathematical concept.

Understanding the Greatest Common Factor (GCF)

Before we delve into the methods, let's solidify our understanding of the GCF. The GCF of two or more integers is the largest number that perfectly divides all the given integers. For example, the GCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 without leaving a remainder. Understanding the GCF is essential in various mathematical applications, including:

Simplifying Fractions: The GCF helps reduce fractions to their simplest form. For instance, simplifying 12/18 requires finding the GCF (which is 6), resulting in the equivalent fraction 2/3.
Solving Algebraic Equations: The GCF plays a vital role in factoring polynomials, a critical step in solving many algebraic equations.
Number Theory: The GCF is a cornerstone concept in number theory, used to explore relationships between integers and solve various number-theoretic problems.

Method 1: Prime Factorization Method

The prime factorization method involves breaking down each number into its prime factors. Prime factors are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.). Once we have the prime factorization of each number, we identify the common prime factors and multiply them to find the GCF.

Let's apply this to find the GCF of 84 and 90:

1. Find the prime factorization of 84:

84 = 2 x 42 = 2 x 2 x 21 = 2 x 2 x 3 x 7 = 2² x 3 x 7

2. Find the 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

3. Identify common prime factors:

Comparing the prime factorizations, we see that both 84 and 90 share a common factor of 2 and a common factor of 3.

4. Multiply the common prime factors:

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

Therefore, the greatest common factor of 84 and 90 is 6.

Method 2: The Euclidean Algorithm

The Euclidean algorithm is an efficient method for finding the GCF, especially 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.

Let's use the Euclidean algorithm to find the GCF of 84 and 90:

1. Start with the larger number (90) and the smaller number (84):

90 = 1 x 84 + 6

2. Replace the larger number (90) with the remainder (6) and repeat the process:

84 = 14 x 6 + 0

3. The process stops when the remainder is 0. The last non-zero remainder is the GCF:

The last non-zero remainder is 6. Therefore, the GCF(84, 90) = 6.

Method 3: Listing Factors Method

This method involves listing all the factors of each number and then identifying the largest common factor. While straightforward for smaller numbers, it can become cumbersome for larger numbers.

1. List the factors of 84:

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

2. List the factors of 90:

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

3. Identify the common factors:

The common factors of 84 and 90 are 1, 2, 3, and 6.

4. Determine the greatest common factor:

The largest common factor is 6. Therefore, the GCF(84, 90) = 6.

Comparing the Methods

Each method offers a different approach to finding the GCF. The prime factorization method provides a clear understanding of the fundamental building blocks of the numbers. The Euclidean algorithm is efficient for larger numbers, offering a systematic way to reduce the problem to smaller, more manageable steps. The listing factors method is simple for smaller numbers but can become impractical for larger integers. The choice of method depends on the context and the size of the numbers involved.

Applications of the GCF

The GCF has numerous applications beyond simply simplifying fractions. It's a crucial concept in various mathematical and real-world scenarios:

Simplifying Ratios: GCF helps express ratios in their simplest form. For instance, a ratio of 84:90 can be simplified to 14:15 by dividing both numbers by their GCF (6).
Dividing Quantities: The GCF helps determine the largest possible equal groups when dividing quantities. If you have 84 apples and 90 oranges, you can create 6 equal groups, each containing 14 apples and 15 oranges.
Geometry Problems: GCF is used in geometry problems related to area, volume, and finding the dimensions of shapes that can be divided equally.
Modular Arithmetic: In modular arithmetic (a system of arithmetic for integers, where numbers "wrap around" upon reaching a given modulus), the GCF plays a crucial role in determining properties like inverses and solutions to congruences.
Cryptography: GCF is used in some cryptographic algorithms, especially those based on the RSA algorithm.

Further Exploration: GCF of More Than Two Numbers

The methods discussed above can be extended to find the GCF of more than two numbers. For the prime factorization method, you would find the prime factorization of each number and then identify the common prime factors across all numbers. For the Euclidean algorithm, you can find the GCF of two numbers, then find the GCF of that result and the next number, and so on. The listing factors method would involve listing the factors of all the numbers and finding the largest factor common to all.

For example, let's find the GCF of 84, 90, and 120:

Prime Factorization:
- 84 = 2² x 3 x 7
- 90 = 2 x 3² x 5
- 120 = 2³ x 3 x 5
The only common prime factor is 2 and 3. Therefore, GCF(84, 90, 120) = 2 x 3 = 6
Euclidean Algorithm (iterative approach):
- GCF(84, 90) = 6 (as calculated before)
- GCF(6, 120) = 6
Therefore, GCF(84, 90, 120) = 6
Listing Factors (This method is less efficient for more than two numbers): This approach would involve listing all factors for each number and identifying the largest common factor among the three sets.

Conclusion

Finding the greatest common factor is a fundamental skill with a wide range of applications in mathematics and beyond. Understanding the different methods – prime factorization, Euclidean algorithm, and listing factors – empowers you to choose the most efficient approach depending on the context and the size of the numbers involved. Mastering this concept strengthens your mathematical foundation and opens doors to more advanced mathematical concepts and real-world problem-solving. The example of finding the GCF of 84 and 90, explored thoroughly in this article, serves as a practical illustration of these fundamental techniques.