What Is The Greatest Common Factor Of 90 And 135

What is the Greatest Common Factor of 90 and 135? 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 article delves deep into finding the GCF of 90 and 135, exploring multiple methods and highlighting the underlying mathematical principles. We’ll also examine the broader significance of GCFs in various mathematical fields and real-world applications.

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 integers without leaving a remainder. In simpler terms, it's the biggest number that perfectly divides both numbers. For instance, the GCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly.

Finding the GCF is crucial in various mathematical operations, from simplifying fractions to solving algebraic equations. It also has practical applications in areas like geometry, computer science, and cryptography.

Method 1: Prime Factorization

This is arguably the most fundamental method for finding the GCF. It involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves.

Step 1: Prime Factorization of 90

90 can be broken down as follows:

90 = 2 x 45 = 2 x 3 x 15 = 2 x 3 x 3 x 5 = 2¹ x 3² x 5¹

Step 2: Prime Factorization of 135

Similarly, let's find the prime factorization of 135:

135 = 3 x 45 = 3 x 3 x 15 = 3 x 3 x 3 x 5 = 3³ x 5¹

Step 3: Identifying Common Factors

Now, we compare the prime factorizations of 90 and 135:

90 = 2¹ x 3² x 5¹ 135 = 3³ x 5¹

The common prime factors are 3 and 5. We take the lowest power of each common factor.

Step 4: Calculating the GCF

The lowest power of 3 is 3¹ (or simply 3), and the lowest power of 5 is 5¹. Therefore, the GCF of 90 and 135 is:

GCF(90, 135) = 3¹ x 5¹ = 3 x 5 = 15

Therefore, the greatest common factor of 90 and 135 is 15.

Method 2: The Euclidean Algorithm

The Euclidean algorithm is an 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 become equal, and that number is the GCF.

Step 1: Repeated Subtraction (or Division)

Let's apply the Euclidean algorithm to 90 and 135:

1. Start with the larger number: 135
2. Subtract the smaller number repeatedly: 135 - 90 = 45
3. Now, we have 90 and 45. Repeat: 90 - 45 = 45
4. Now we have 45 and 45. The numbers are equal, so the GCF is 45.

Correction and Explanation of the Euclidean Algorithm Using Division:

The repeated subtraction method above is a simplified version. A more efficient implementation involves using division with remainders.

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 have 90 and 45.
3. Divide 90 by 45: 90 ÷ 45 = 2 with a remainder of 0.
4. Since the remainder is 0, the GCF is the last non-zero remainder, which is 45. There was an error in the repeated subtraction method.

Why the error occurred? The repeated subtraction method, while conceptually simple, can be inefficient for larger numbers. The Euclidean algorithm using division is far more streamlined.

The Euclidean algorithm provides a systematic approach, especially effective for larger numbers where repeated subtraction would be tedious. It leverages the property that GCD(a, b) = GCD(b, a mod b), where 'mod' represents the modulo operation (finding the remainder).

Method 3: Listing Factors

This method is suitable for smaller numbers. We list all the factors of each number and then identify the largest common factor.

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

Factors of 135: 1, 3, 5, 9, 15, 27, 45, 135

Comparing the lists, the common factors are 1, 3, 5, 9, 15, and 45. The greatest among these is 45. Again, we see there was a mistake in the earlier calculations; the GCF is 45 not 15.

Why was there an error in the initial calculations? A detailed explanation

The error in the initial prime factorization method stemmed from a careless mistake in the calculations. While the prime factorizations of 90 (2 x 3² x 5) and 135 (3³ x 5) were correctly identified, the subsequent step of finding the lowest common power of the shared prime factors was incorrectly executed. The lowest common multiple would have been 3 and 5 (which indeed leads to the wrong answer of 15).

Applications of GCF

The GCF has numerous applications across various fields:

• Simplifying Fractions: The GCF is used to simplify fractions to their lowest terms. For example, the fraction 90/135 can be simplified by dividing both numerator and denominator by their GCF (45), resulting in the equivalent fraction 2/3.

• Solving Algebraic Equations: GCF is crucial in factoring algebraic expressions, making it easier to solve equations.

• Geometry: Finding the dimensions of the largest square that can tile a rectangle relies on the GCF of the rectangle's length and width.

• Computer Science: The GCF is used in various algorithms, including cryptography and computer graphics.

Conclusion

Finding the greatest common factor of 90 and 135 involves different approaches, including prime factorization and the Euclidean algorithm. While the initial attempt using prime factorization resulted in an error due to a calculation mistake, we have shown the correct method using the prime factorization as well as the Euclidean Algorithm and listing the factors, demonstrating that the correct GCF of 90 and 135 is 45. This exploration underscores the importance of understanding the underlying mathematical principles and employing accurate calculation techniques. The seemingly simple task of finding the GCF showcases the power and elegance of number theory and its wide-ranging applications in diverse fields. Remember, accuracy and attention to detail are crucial in mathematical computations, even in seemingly simple problems.