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) or highest common factor (HCF), is the largest positive integer that divides each of the integers without leaving a remainder. Finding the GCF is a fundamental concept in mathematics with applications ranging from simplifying fractions to solving algebraic equations. This article will explore multiple methods for determining the GCF of 84 and 90, providing a thorough understanding of the process and its underlying principles.

Understanding Prime Factorization

Before diving into the methods, let's establish a crucial concept: prime factorization. Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are whole numbers greater than 1 that have only two divisors: 1 and themselves (e.g., 2, 3, 5, 7, 11).

Prime factorization provides a fundamental building block for finding the GCF. By breaking down numbers into their prime factors, we can easily identify the common factors and determine the greatest one.

Prime Factorization of 84

Let's start by finding the prime factorization of 84:

• Divide by 2: 84 ÷ 2 = 42
• Divide by 2 again: 42 ÷ 2 = 21
• Divide by 3: 21 ÷ 3 = 7
• 7 is a prime number: The factorization ends here.

Therefore, the prime factorization of 84 is 2 x 2 x 3 x 7, or 2² x 3 x 7.

Prime Factorization of 90

Now, let's find the prime factorization of 90:

• Divide by 2: 90 ÷ 2 = 45
• Divide by 3: 45 ÷ 3 = 15
• Divide by 3 again: 15 ÷ 3 = 5
• 5 is a prime number: The factorization ends here.

Therefore, the prime factorization of 90 is 2 x 3 x 3 x 5, or 2 x 3² x 5.

Method 1: Using Prime Factorization to Find the GCF

Once we have the prime factorization of both numbers, finding the GCF becomes straightforward. We identify the common prime factors and multiply them together.

Comparing the prime factorizations of 84 (2² x 3 x 7) and 90 (2 x 3² x 5), we see that they share only 2 and 3 as common prime factors.

The lowest power of the common prime factor 2 is 2¹. The lowest power of the common prime factor 3 is 3¹.

Therefore, the GCF of 84 and 90 is 2 x 3 = 6.

Method 2: The Euclidean Algorithm

The Euclidean algorithm provides an efficient method for finding the GCF, particularly for larger numbers where prime factorization might be more tedious. This algorithm relies on repeated application of the division algorithm.

The Euclidean 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. This process is repeated until the two numbers are equal, and that number is the GCF. A more efficient version uses the modulo operation (remainder after division).

Let's apply the Euclidean algorithm to find the GCF of 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: Now we find the GCF of 84 and 6.
3. Divide the larger number (84) by the smaller number (6): 84 ÷ 6 = 14 with a remainder of 0.
4. Since the remainder is 0, the GCF is the last non-zero remainder, which is 6.

Therefore, the GCF of 84 and 90 using the Euclidean algorithm is 6.

Method 3: Listing Factors

This method is suitable for smaller numbers. We list all the factors of each number and identify 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 two lists, we find the common factors: 1, 2, 3, and 6. The largest common factor is 6.

Applications of Finding the GCF

Finding the greatest common factor has numerous applications in various mathematical contexts:

• Simplifying Fractions: The GCF is used to simplify fractions to their lowest terms. For example, the fraction 84/90 can be simplified by dividing both the numerator and denominator by their GCF, which is 6, resulting in the simplified fraction 14/15.

• Solving Algebraic Equations: The GCF plays a role in factoring algebraic expressions. Finding the GCF of the terms in an expression allows for simplification and solving equations.

• Number Theory: The GCF is a fundamental concept in number theory, used in various theorems and proofs.

• Geometry: GCF is used in solving geometric problems involving areas, perimeters, and other related concepts.

• Real-World Applications: The concept of GCF finds applications in various real-world scenarios, such as dividing items into equal groups or determining the size of the largest square tile that can be used to cover a rectangular area.

Comparing the Methods

Each method for finding the GCF has its strengths and weaknesses:

• Prime Factorization: This method is conceptually straightforward and easy to understand, but it can become time-consuming for large numbers whose prime factorization is difficult to find.

• Euclidean Algorithm: This is the most efficient method, especially for larger numbers, as it avoids the need for complete prime factorization.

• Listing Factors: This is the simplest method for small numbers but becomes impractical for larger numbers due to the increasing number of factors to consider.

Conclusion

Finding the greatest common factor (GCF) of 84 and 90, whether through prime factorization, the Euclidean algorithm, or listing factors, consistently yields the result of 6. Understanding these different methods provides a versatile toolkit for tackling GCF problems across various mathematical contexts. Choosing the most appropriate method depends on the size of the numbers involved and the level of mathematical proficiency required. The ability to find the GCF is a crucial skill in mathematics with wide-ranging applications beyond theoretical concepts. Mastering these techniques strengthens foundational mathematical understanding and opens doors to tackling more complex problems.