What Is The Greatest Common Factor Of 75 And 90

What is the Greatest Common Factor of 75 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 concept with surprisingly deep roots in number theory and practical applications across various fields. This comprehensive guide will not only answer the question, "What is the greatest common factor of 75 and 90?" but also explore the underlying principles, different methods of calculation, and the significance of GCF in mathematics and beyond.

Understanding the Greatest Common Factor (GCF)

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides each of the two or more integers without leaving a remainder. It's essentially the biggest number that perfectly divides both numbers. Think of it as the largest shared building block of the numbers.

For example, 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.

Finding the GCF of 75 and 90: Three Proven Methods

Now, let's tackle the central question: What is the greatest common factor of 75 and 90? We'll explore three common methods:

1. Listing Factors Method

This is the most straightforward approach, particularly useful for smaller numbers. We'll list all the factors of each number and then identify the largest factor they have in common.

• Factors of 75: 1, 3, 5, 15, 25, 75
• Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Comparing the lists, we see that the common factors are 1, 3, 5, and 15. The greatest among these is 15. Therefore, the GCF of 75 and 90 is 15.

2. Prime Factorization Method

This method uses the prime factorization of each number. The prime factorization of a number is expressing it as a product of prime numbers (numbers only divisible by 1 and themselves).

• Prime factorization of 75: 3 x 5 x 5 = 3 x 5²
• Prime factorization of 90: 2 x 3 x 3 x 5 = 2 x 3² x 5

To find the GCF, we identify the common prime factors and take the lowest power of each. Both 75 and 90 share a 3 and a 5. The lowest power of 3 is 3¹ (or just 3), and the lowest power of 5 is 5¹. Multiplying these together, we get 3 x 5 = 15. Thus, the GCF is 15.

3. Euclidean Algorithm

This is a more efficient method for larger numbers. The Euclidean algorithm is 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.

1. Start with the two numbers: 75 and 90.
2. Subtract the smaller number from the larger: 90 - 75 = 15
3. Replace the larger number with the result: Now we have 15 and 75.
4. Repeat the subtraction: 75 - 15 = 60
5. Now we have 15 and 60.
6. Repeat: 60 - 15 = 45
7. Now we have 15 and 45.
8. Repeat: 45 - 15 = 30
9. Now we have 15 and 30. 10.Repeat: 30 - 15 = 15 11.Now we have 15 and 15. The numbers are equal, so the GCF is 15.

This iterative process quickly leads to the GCF, making it particularly useful for larger numbers where listing factors becomes cumbersome. The Euclidean algorithm can also be expressed using a modulo operation (%). The remainder after division becomes the new smaller number until the remainder is 0, and the last non-zero remainder is the GCF.

Applications of the Greatest Common Factor

The GCF isn't just an abstract mathematical concept; it has practical applications in various fields:

• Simplifying Fractions: The GCF is crucial for simplifying fractions to their lowest terms. To simplify a fraction, divide both the numerator and the denominator by their GCF. For instance, the fraction 75/90 can be simplified to 5/6 by dividing both by their GCF, 15.

• Geometry: GCF is used in solving geometric problems involving area and perimeter. For example, finding the largest possible square tiles to cover a rectangular floor requires finding the GCF of the length and width of the floor.

• Cryptography: The GCF plays a significant role in cryptography, particularly in the RSA algorithm, a widely used public-key cryptosystem. The algorithm relies heavily on the properties of prime numbers and their GCF.

• Scheduling: Determining the least common multiple (LCM) – the smallest number that is a multiple of both numbers – often involves finding the GCF first. LCM and GCF are used in scheduling problems, such as determining when events will occur simultaneously.

• Music: GCF and LCM are utilized in music theory to determine harmonious intervals and understand relationships between different musical notes.

Beyond the Basics: Exploring Related Concepts

Understanding the GCF opens doors to exploring related mathematical concepts:

• Least Common Multiple (LCM): The LCM is the smallest positive integer that is a multiple of both numbers. The relationship between GCF and LCM is given by the formula: GCF(a, b) x LCM(a, b) = a x b.

• Modular Arithmetic: Modular arithmetic involves working with remainders after division. The GCF plays a critical role in solving congruences and understanding properties of modular arithmetic.

• Diophantine Equations: These equations involve finding integer solutions. The GCF is often used in determining the solvability and finding solutions to Diophantine equations.

• Number Theory: The GCF is a fundamental concept in number theory, the branch of mathematics that studies the properties of integers.

Conclusion: The Power of a Simple Concept

The seemingly simple task of finding the greatest common factor of 75 and 90 reveals a rich tapestry of mathematical concepts with wide-ranging applications. Whether you're simplifying fractions, solving geometric problems, or delving into the intricacies of cryptography, understanding the GCF is a valuable asset. The various methods discussed – listing factors, prime factorization, and the Euclidean algorithm – provide versatile approaches to determining the GCF, each with its own strengths and weaknesses depending on the numbers involved. The journey to find the GCF of 75 and 90 is, in essence, a journey into the heart of number theory and its powerful applications in the world around us. The answer, as we've repeatedly shown, remains consistently 15.