What Is The Greatest Common Factor Of 84 And 90

What is the Greatest Common Factor of 84 and 90? A Deep Dive into Number Theory

Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers might seem like a simple arithmetic task. However, understanding the underlying principles and various methods for calculating the GCF opens a fascinating window into number theory and its applications in diverse fields. This article will not only answer the question of what the GCF of 84 and 90 is but also explore different approaches to finding the GCF, their underlying mathematical concepts, and practical implications.

Understanding Greatest Common Factor (GCF)

The greatest common factor (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 can perfectly divide both numbers. This concept is fundamental in simplifying fractions, solving algebraic equations, and even in advanced areas like cryptography.

Key Concepts:

• Factors: A factor of a number is a whole number that divides the number evenly (without a remainder). For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12.
• Common Factors: Common factors are the factors that two or more numbers share. For example, the common factors of 12 and 18 are 1, 2, 3, and 6.
• Greatest Common Factor (GCF): The largest of these common factors is the GCF. In the case of 12 and 18, the GCF is 6.

Methods for Finding the GCF of 84 and 90

Let's delve into various techniques to determine the GCF of 84 and 90. We'll explore the most common methods, explaining each step clearly.

1. Listing Factors Method

This is a straightforward approach, especially useful for smaller numbers. We list all the factors of each number and then 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

Common Factors: 1, 2, 3, 6

GCF: 6

This method is simple but can become cumbersome with larger numbers.

2. Prime Factorization Method

This method uses the fundamental theorem of arithmetic, which states that every integer greater than 1 can be uniquely represented as a product of prime numbers (numbers divisible only by 1 and themselves).

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

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

Now, we identify the common prime factors and their lowest powers:

• Common prime factor: 2 (with power 1)
• Common prime factor: 3 (with power 1)

GCF = 2¹ x 3¹ = 6

3. Euclidean Algorithm

This is a highly efficient method for finding the GCF, especially for larger numbers. It's 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.

Steps:

1. Divide the larger number (90) by the smaller number (84) and find the remainder. 90 ÷ 84 = 1 with a remainder of 6.

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

Applications of GCF

The GCF finds practical applications in various fields:

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

• Algebra: GCF plays a crucial role in factoring algebraic expressions. Finding the GCF of the terms in an expression allows for simplification and solving equations.

• Geometry: GCF is used in problems related to finding the dimensions of squares or rectangles that can be formed from a given area.

• Cryptography: Concepts related to GCF, particularly the Euclidean algorithm, are fundamental to many cryptographic algorithms.

Beyond the Basics: Exploring Further Concepts

Understanding the GCF leads to a deeper appreciation of more advanced number theory concepts:

• Least Common Multiple (LCM): The LCM is the smallest positive integer that is a multiple of two or more numbers. The GCF and LCM are related by the formula: GCF(a, b) x LCM(a, b) = a x b

• Modular Arithmetic: This branch of number theory deals with remainders after division. The GCF plays a vital role in understanding modular arithmetic and its applications.

• Diophantine Equations: These equations involve finding integer solutions to algebraic equations. The GCF is often a key element in determining the solvability and solutions of Diophantine equations.

Conclusion: The Power of the GCF

The seemingly simple concept of the greatest common factor opens doors to a rich landscape of mathematical ideas and practical applications. By understanding the different methods for calculating the GCF and its connections to other mathematical concepts, we gain a deeper appreciation for the elegance and power of number theory. While the GCF of 84 and 90 is definitively 6, the journey to find it and the broader understanding it unlocks is far more significant. This exploration highlights the importance of fundamental mathematical principles and their surprisingly wide-ranging impact across various disciplines. Further exploration into these related areas will continue to enrich your mathematical knowledge and problem-solving skills.