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 with applications in various fields, from simplifying fractions to solving algebraic problems. This comprehensive guide will explore multiple methods for determining the GCF of 84 and 90, providing a clear understanding of the underlying principles and practical applications.

Understanding the Concept of Greatest Common Factor

Before diving into the methods, let's solidify our understanding of the GCF. Consider two numbers, 'a' and 'b'. The GCF of 'a' and 'b' is the largest number that perfectly divides both 'a' and 'b'. For instance, 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. Therefore, the greatest common factor of 12 and 18 is 6.

This concept extends to any number of integers. Finding the GCF is crucial for simplifying fractions to their lowest terms. For example, simplifying the fraction 12/18 requires finding the GCF (which is 6), allowing us to simplify the fraction to 2/3.

Method 1: Prime Factorization

This is a widely used and reliable method for finding the GCF. It involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves.

Step 1: Prime Factorization of 84

84 can be broken down as follows:

• 84 = 2 x 42
• 84 = 2 x 2 x 21
• 84 = 2 x 2 x 3 x 7
• Therefore, the prime factorization of 84 is 2² x 3 x 7

Step 2: Prime Factorization of 90

90 can be broken down as follows:

• 90 = 2 x 45
• 90 = 2 x 3 x 15
• 90 = 2 x 3 x 3 x 5
• Therefore, the prime factorization of 90 is 2 x 3² x 5

Step 3: Identifying Common Factors

Now, compare the prime factorizations of 84 (2² x 3 x 7) and 90 (2 x 3² x 5). The common prime factors are 2 and 3.

Step 4: Calculating the GCF

To find the GCF, take the lowest power of each common prime factor and multiply them together. In this case:

• The lowest power of 2 is 2¹ (or simply 2).
• The lowest power of 3 is 3¹.

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

Method 2: Listing Factors

This method is straightforward but can be less efficient for larger numbers.

Step 1: List all factors of 84

The factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Step 2: List all factors of 90

The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Step 3: Identify Common Factors

Compare the two lists and identify the common factors: 1, 2, 3, 6.

Step 4: Determine the GCF

The greatest common factor from the list is 6. Therefore, the GCF of 84 and 90 is 6.

Method 3: Euclidean Algorithm

The Euclidean algorithm is an efficient method, 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.

Step 1: Apply the Algorithm

1. Start with the two numbers: 84 and 90.
2. Subtract the smaller number (84) from the larger number (90): 90 - 84 = 6
3. Replace the larger number with the result (6), keeping the smaller number (84). Now we have 6 and 84.
4. Repeat: 84 - 6 x 14 = 0. The remainder is 0.

Step 2: Determine the GCF

The last non-zero remainder is the GCF. In this case, the last non-zero remainder before reaching 0 is 6. Therefore, the GCF of 84 and 90 is 6.

Applications of Finding the GCF

The GCF has numerous applications across different areas of mathematics and beyond:

• Simplifying Fractions: As mentioned earlier, finding the GCF allows us to reduce fractions to their simplest form.
• Solving Algebraic Equations: The GCF is used in factoring polynomials, which is crucial for solving many algebraic equations.
• Geometry: The GCF can be used to determine the dimensions of the largest square that can tile a rectangle of given dimensions.
• Number Theory: The GCF plays a significant role in various concepts within number theory, such as modular arithmetic and Diophantine equations.
• Computer Science: Algorithms for finding the GCF are used in cryptography and other computational tasks.

Choosing the Right Method

The best method for finding the GCF depends on the numbers involved and your familiarity with each method.

• Prime Factorization: This is generally preferred for smaller numbers as it provides a clear understanding of the factors involved. It can become tedious with very large numbers.
• Listing Factors: Suitable only for relatively small numbers due to the potential for lengthy factor lists.
• Euclidean Algorithm: This is the most efficient method for larger numbers, particularly when dealing with numbers that are difficult to factorize directly.

Conclusion: Mastering the GCF

Understanding and applying different methods for calculating the greatest common factor is essential for various mathematical applications. The prime factorization method provides a fundamental understanding, while the Euclidean algorithm offers efficiency for larger numbers. Mastering these techniques empowers you to confidently tackle problems involving GCFs and simplifies many complex mathematical computations. The GCF of 84 and 90, as demonstrated through each method, is definitively 6. This foundational concept unlocks a deeper understanding of number theory and its practical applications in numerous fields. Remember to choose the most appropriate method based on the context and the size of the numbers involved.