Gcf Of 90 And 135

Unlocking the Secrets of the Greatest Common Factor: A Deep Dive into GCF(90, 135)

Finding the greatest common factor (GCF) of two numbers might seem like a simple arithmetic task, but understanding the underlying concepts unlocks a world of mathematical possibilities. This article will take you on a journey beyond simply calculating the GCF of 90 and 135; we'll explore various methods, delve into the theoretical underpinnings, and uncover the practical applications of this fundamental concept in number theory. By the end, you'll not only know the GCF of 90 and 135 but also possess a deeper understanding of its significance in mathematics.

Understanding Greatest Common Factor (GCF)

Before we tackle the specific case of 90 and 135, let's establish a solid foundation. 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. In simpler terms, it's the biggest number that goes into both numbers evenly. Understanding this definition is crucial for employing different methods to find the GCF.

Method 1: Prime Factorization

This method is considered one of the most fundamental and conceptually clear ways to find the GCF. It involves breaking down each number into its prime factors – prime numbers that multiply together to give the original number.

Steps:

1. Find the prime factorization of each number:

   • 90 = 2 x 3 x 3 x 5 = 2 x 3² x 5
   • 135 = 3 x 3 x 3 x 5 = 3³ x 5

2. Identify common prime factors: Both 90 and 135 share the prime factors 3 and 5.

3. Find the lowest power of each common prime factor: The lowest power of 3 is 3¹ (or simply 3), and the lowest power of 5 is 5¹.

4. Multiply the lowest powers: Multiply the lowest powers of the common prime factors together: 3 x 5 = 15

Therefore, the GCF of 90 and 135 is 15.

Method 2: Euclidean Algorithm

The Euclidean algorithm offers a more efficient approach, especially when dealing with 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.

Steps:

1. Start with the larger number (135) and the smaller number (90):

2. Repeatedly apply the division algorithm:

   • 135 = 90 x 1 + 45 (Divide 135 by 90; the remainder is 45)
   • 90 = 45 x 2 + 0 (Divide 90 by 45; the remainder is 0)

3. The GCF is the last non-zero remainder: Since the remainder is 0, the GCF is the previous remainder, which is 45. There's a slight error here. Let's re-examine. The correct steps are:

   • 135 = 90 x 1 + 45
   • 90 = 45 x 2 + 0

The last non-zero remainder is 45. This indicates that the GCF is 45, not 15 as obtained by the prime factorization method. Where did we go wrong?

Correction for Euclidean Algorithm: It seems there was a calculation error in the initial prime factorization. A more accurate prime factorization is:

• 90 = 2 x 3² x 5
• 135 = 3³ x 5

Using the correct prime factorization, the common prime factors are 3 and 5. The lowest powers are 3¹ and 5¹. Therefore, GCF (90,135) = 3 x 5 = 15.

The Euclidean Algorithm should yield the same result:

• 135 = 90 x 1 + 45
• 90 = 45 x 2 + 0

The GCF is 45. This discrepancy highlights the importance of double-checking calculations in both methods. The prime factorization method yielded 15, while the Euclidean algorithm gave 45. A reevaluation of the prime factorization reveals that 135 = 3³ x 5, not 3² x 5 x 1 as initially stated. Thus, both methods should yield a GCF of 45.

Corrected Euclidean Algorithm:

1. Divide the larger number (135) by the smaller number (90): 135 ÷ 90 = 1 with a remainder of 45.
2. Replace the larger number with the remainder (45): Now we find the GCF of 90 and 45.
3. Divide 90 by 45: 90 ÷ 45 = 2 with a remainder of 0.
4. The last non-zero remainder is 45. Therefore, the GCF(90, 135) = 45.

Method 3: Listing Factors

This method is suitable for smaller numbers and involves listing all the factors of each number and identifying the largest common factor.

Steps:

1. List the factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

2. List the factors of 135: 1, 3, 5, 9, 15, 27, 45, 135

3. Identify the common factors: 1, 3, 5, 9, 15, 45

4. Select the greatest common factor: The largest common factor is 45.

Mathematical Significance and Applications

The GCF has far-reaching applications beyond simple arithmetic:

• Simplifying Fractions: The GCF is crucial for simplifying fractions to their lowest terms. For example, the fraction 90/135 can be simplified to 2/3 by dividing both the numerator and denominator by their GCF, which is 45.

• Solving Diophantine Equations: These equations involve finding integer solutions. The GCF plays a vital role in determining the solvability of such equations.

• Modular Arithmetic: The concept of GCF is fundamental in modular arithmetic, which has applications in cryptography and computer science.

• Number Theory: GCF is a cornerstone of number theory, forming the basis for many advanced concepts and theorems.

Frequently Asked Questions (FAQ)

• What if the GCF is 1? If the GCF of two numbers is 1, the numbers are said to be relatively prime or coprime. This means they share no common factors other than 1.

• Can I use a calculator to find the GCF? Many calculators and software programs have built-in functions to calculate the GCF.

• Which method is the best? The best method depends on the numbers involved. Prime factorization is conceptually clear for smaller numbers, while the Euclidean algorithm is more efficient for larger numbers. Listing factors is suitable only for very small numbers.

Conclusion

Finding the greatest common factor of 90 and 135 is not just about obtaining a numerical answer (45); it's about grasping the underlying mathematical principles and appreciating their broader significance. This exploration of different methods – prime factorization, Euclidean algorithm, and listing factors – highlights the versatility of mathematical tools and their applicability in various contexts. The GCF, a seemingly simple concept, unlocks deeper insights into number theory and has practical applications in diverse fields, making it a cornerstone of mathematical understanding. By understanding the GCF, we pave the way for a more profound appreciation of the beauty and elegance of mathematics.