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) of two numbers might seem like a simple arithmetic task, but it's a fundamental concept in number theory with applications far beyond basic math. This comprehensive guide will not only answer the question "What is the greatest common factor of 84 and 90?" but also explore various methods to find the GCF, delve into its importance, and showcase its real-world applications.

Understanding the Greatest Common Factor (GCF)

The greatest common factor (GCF), also known as the greatest common divisor (GCD), 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. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 evenly.

Understanding the GCF is crucial for simplifying fractions, solving algebraic equations, and even in more advanced mathematical fields like cryptography.

Methods for Finding the GCF of 84 and 90

Several methods can be used to determine the GCF of 84 and 90. Let's explore the most common ones:

1. Listing Factors Method

This method involves listing all the factors of each number and then identifying 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 lists, we can see that the common factors are 1, 2, 3, and 6. The greatest common factor is therefore 6.

2. Prime Factorization Method

This is a more efficient method, especially for larger numbers. It involves finding the prime factorization of each number and then multiplying the common prime factors raised to the lowest power.

Prime factorization of 84: 2² x 3 x 7 Prime factorization of 90: 2 x 3² x 5

The common prime factors are 2 and 3. The lowest power of 2 is 2¹ (or 2), and the lowest power of 3 is 3¹. Therefore, the GCF is 2 x 3 = 6.

3. Euclidean Algorithm

The Euclidean algorithm 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 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 apply the Euclidean algorithm to 84 and 90:

1. 90 = 84 x 1 + 6 (Subtract 84 from 90, the remainder is 6)
2. 84 = 6 x 14 + 0 (Divide 84 by 6, the remainder is 0)

Since the remainder is 0, the GCF is the last non-zero remainder, which is 6.

Why is Finding the GCF Important?

The GCF has numerous applications in various fields:

1. Simplifying Fractions

The GCF is essential for simplifying fractions to their lowest terms. To simplify a fraction, divide both the numerator and the denominator by their GCF. For example, to simplify the fraction 84/90, we divide both the numerator and denominator by their GCF, which is 6:

84/90 = (84 ÷ 6) / (90 ÷ 6) = 14/15

2. Solving Algebraic Equations

The GCF plays a role in factoring algebraic expressions. Factoring an expression involves finding the GCF of the terms and then rewriting the expression as a product of the GCF and the remaining factors. This simplifies equations and makes them easier to solve.

3. Geometry and Measurement

The GCF is used in problems involving lengths, areas, and volumes. For instance, finding the largest square tile that can perfectly cover a rectangular floor requires finding the GCF of the length and width of the floor.

4. Cryptography

The GCF, particularly the Euclidean algorithm for finding it, is a fundamental component in certain cryptographic algorithms. These algorithms are used to secure online communications and data.

5. Number Theory and Abstract Algebra

The GCF forms the basis for many concepts in number theory and abstract algebra, including modular arithmetic, which has applications in computer science and cryptography.

Beyond the Basics: Exploring Related Concepts

Understanding the GCF opens the door to other important concepts in number theory:

1. Least Common Multiple (LCM)

The least common multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of all the integers. The LCM and GCF are related by the following formula:

LCM(a, b) x GCF(a, b) = a x b

For 84 and 90, the LCM is 1260. You can verify this using the formula: 6 (GCF) x 1260 (LCM) = 7560, and 84 x 90 = 7560.

2. Relatively Prime Numbers

Two numbers are considered relatively prime (or coprime) if their GCF is 1. For example, 15 and 28 are relatively prime because their GCF is 1.

3. Modular Arithmetic

Modular arithmetic involves performing arithmetic operations within a specific range (the modulus). The GCF plays a crucial role in determining the existence and uniqueness of solutions in modular arithmetic equations.

Real-World Applications of the GCF

The GCF finds its way into many real-world scenarios:

• Dividing items equally: Imagine you have 84 apples and 90 oranges, and you want to divide them into identical bags, with the same number of apples and oranges in each bag. The GCF (6) tells you that you can make 6 identical bags, each containing 14 apples and 15 oranges.

• Construction and design: The GCF is essential in calculating the dimensions of tiles or bricks needed to cover a surface perfectly.

• Music theory: The GCF is used in determining musical intervals and harmonies.

• Computer science: The GCF and the Euclidean algorithm are crucial in various computer algorithms, including those used in cryptography and data compression.

Conclusion

Finding the greatest common factor of 84 and 90, which is 6, is more than a simple arithmetic exercise. It's a fundamental concept with far-reaching implications in mathematics, computer science, and various other fields. Understanding the GCF and the different methods for finding it provides a solid foundation for tackling more complex mathematical problems and real-world applications. By mastering these techniques, you unlock a deeper understanding of the underlying principles of number theory and its practical utility. From simplifying fractions to securing online transactions, the GCF quietly but powerfully shapes our world.