Greatest Common Factor Of 84 And 105

Finding the Greatest Common Factor (GCF) of 84 and 105: A Comprehensive Guide

Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics with applications ranging from simplifying fractions to solving algebraic equations. This comprehensive guide will explore various methods to determine the GCF of 84 and 105, delve into the underlying mathematical principles, and provide practical examples to solidify your understanding.

Understanding Greatest Common Factor (GCF)

Before we dive into calculating the GCF of 84 and 105, let's clarify what it means. The GCF of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. In simpler terms, it's the biggest number that perfectly divides both numbers. Understanding this definition is crucial to grasping the various methods for finding the GCF.

Method 1: Prime Factorization

This method is considered a classic and reliable approach. It involves breaking down each number into its prime factors and then identifying the common factors to find the GCF.

Step-by-Step Guide for Prime Factorization:

1. Find the prime factorization of 84:

   84 can be broken down as follows:

   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 105:

   105 can be broken down as follows:

   105 = 3 x 35 = 3 x 5 x 7

3. Identify common prime factors:

   Both 84 and 105 share the prime factors 3 and 7.

4. Calculate the GCF:

   To find the GCF, multiply the common prime factors together:

   GCF(84, 105) = 3 x 7 = 21

Therefore, the greatest common factor of 84 and 105 is 21. This means 21 is the largest number that divides both 84 and 105 without leaving a remainder.

Method 2: Listing Factors

This method is simpler for smaller numbers but can become cumbersome for larger numbers. It involves listing all the factors of each number and then identifying the largest common factor.

Step-by-Step Guide for Listing Factors:

1. List the factors of 84:

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

2. List the factors of 105:

   Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

3. Identify common factors:

   The common factors of 84 and 105 are 1, 3, 7, and 21.

4. Determine the GCF:

   The largest common factor is 21.

Therefore, the GCF(84, 105) = 21, confirming the result obtained using prime factorization.

Method 3: Euclidean Algorithm

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

Step-by-Step Guide for Euclidean Algorithm:

1. Start with the two numbers:

   84 and 105

2. Divide the larger number by the smaller number and find the remainder:

   105 ÷ 84 = 1 with a remainder of 21

3. Replace the larger number with the smaller number and the smaller number with the remainder:

   New numbers: 84 and 21

4. Repeat the process:

   84 ÷ 21 = 4 with a remainder of 0

5. The GCF is the last non-zero remainder:

   The last non-zero remainder is 21.

Therefore, the GCF(84, 105) = 21, demonstrating the efficiency of the Euclidean algorithm.

Applications of GCF

The GCF has several practical applications in various fields:

• Simplifying Fractions: The GCF is used to simplify fractions to their lowest terms. For example, the fraction 84/105 can be simplified by dividing both the numerator and the denominator by their GCF (21), resulting in the simplified fraction 4/5.

• Solving Equations: GCF plays a role in solving Diophantine equations, which are algebraic equations where only integer solutions are sought.

• Geometry: GCF is used in geometry problems involving finding the dimensions of objects with common factors. For instance, finding the largest square tile that can perfectly cover a rectangular floor with dimensions 84 units and 105 units.

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 example, to find the GCF of 84, 105, and another number, say 147:

1. Find the prime factorization of each number:

   • 84 = 2² x 3 x 7
   • 105 = 3 x 5 x 7
   • 147 = 3 x 7²

2. Identify common prime factors: The common prime factors are 3 and 7.

3. Calculate the GCF: The GCF(84, 105, 147) = 3 x 7 = 21.

Conclusion

Determining the greatest common factor is a fundamental skill in mathematics with diverse applications. This guide has explored three effective methods – prime factorization, listing factors, and the Euclidean algorithm – providing a comprehensive understanding of how to calculate the GCF of 84 and 105. The examples provided illustrate the practical uses of GCF in simplifying fractions and other mathematical problems. Mastering these techniques will enhance your problem-solving capabilities and provide a solid foundation for more advanced mathematical concepts. Remember to choose the method that best suits the numbers involved; for smaller numbers, listing factors might be quicker, while for larger numbers, the Euclidean algorithm proves highly efficient. Understanding the underlying principles of GCF will not only improve your mathematical skills but also strengthen your analytical thinking.