Greatest Common Factor Of 90 And 135

Finding the Greatest Common Factor (GCF) of 90 and 135: A Comprehensive Guide

Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics with applications spanning various fields, from simplifying fractions to solving algebraic equations. This article delves deep into the process of determining the GCF of 90 and 135, exploring multiple methods and highlighting the importance of understanding this concept. We'll also touch upon the practical applications of finding the GCF.

Understanding the Greatest Common Factor (GCF)

Before we jump into calculating the GCF of 90 and 135, let's solidify our understanding of what a GCF actually is. The GCF of two or more numbers is the largest number that divides evenly into all of them without leaving a remainder. Think of it as the biggest common "chunk" you can divide both numbers into.

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. The greatest of these common factors is 6, therefore, the GCF of 12 and 18 is 6.

Method 1: Listing Factors

The most straightforward method to find the GCF, especially for smaller numbers like 90 and 135, is to 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

Identifying the GCF:

Comparing the two lists, we can see that the common factors are 1, 3, 5, 9, 15, and 45. The greatest of these common factors is 45. Therefore, the GCF of 90 and 135 is 45.

This method is simple to understand and effective for smaller numbers, but it can become cumbersome and time-consuming when dealing with larger numbers.

Method 2: Prime Factorization

Prime factorization is a more efficient and systematic method for finding the GCF, especially when dealing with larger numbers. This method involves expressing each number as a product of its prime factors. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11...).

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

Prime Factorization of 135:

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

Finding the GCF using Prime Factorization:

Once we have the prime factorization of both numbers, we identify the common prime factors and their lowest powers.

Both 90 and 135 share the prime factors 3 and 5.

• The lowest power of 3 is 3¹ (or simply 3).
• The lowest power of 5 is 5¹.

Multiplying these common prime factors raised to their lowest powers gives us the GCF: 3 x 5 = 15.

Important Note: There seems to be a discrepancy between the result obtained using the listing factors method (45) and the prime factorization method (15). Let's investigate this further. It appears there's an error in the initial prime factorization of 90. Let's correct it:

Corrected Prime Factorization of 90:

90 = 2 x 3² x 5

Corrected Prime Factorization of 135:

135 = 3³ x 5

Now, let's find the GCF again using the corrected prime factorizations. Both numbers share 3 and 5 as prime factors. The lowest power of 3 is 3², and the lowest power of 5 is 5¹. Therefore, the GCF is 3² x 5 = 9 x 5 = 45. This matches the result from the listing factors method.

Method 3: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF of two numbers, 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, which is the GCF.

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

1. Start with the larger number (135) and the smaller number (90).
2. Subtract the smaller number from the larger number: 135 - 90 = 45
3. Now, we have the numbers 90 and 45.
4. Repeat the process: 90 - 45 = 45
5. We now have 45 and 45. Since both numbers are the same, the GCF is 45.

The Euclidean algorithm is remarkably efficient, requiring fewer steps than the prime factorization method for larger numbers.

Applications of the Greatest Common Factor

The GCF is not just a mathematical curiosity; it has numerous 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 denominator by their GCF. For example, to simplify the fraction 90/135, we divide both by their GCF, 45, resulting in the simplified fraction 2/3.

• Algebra: GCF is essential in factoring algebraic expressions. Finding the GCF of the terms in an expression allows us to simplify and solve equations more efficiently.

• Geometry: The GCF can be used to determine the dimensions of the largest possible square that can tile a rectangular area.

• Measurement and Division: When working with different units of measurement or dividing quantities, determining the GCF helps to find the largest common unit or the largest number of items that can be divided evenly.

• Computer Science: The GCF plays a vital role in algorithms used in cryptography and data compression.

Conclusion

Finding the greatest common factor (GCF) is a fundamental mathematical skill with wide-ranging applications. While listing factors is a simple approach for smaller numbers, prime factorization and the Euclidean algorithm offer more efficient methods for larger numbers. Understanding the concept of GCF and mastering different calculation methods is crucial for success in various mathematical and practical contexts. By employing these techniques, you'll be equipped to tackle GCF problems with confidence and efficiency. Remember that accuracy is paramount, as demonstrated by the initial error in the prime factorization of 90, highlighting the importance of careful calculation and verification. The final, correct GCF of 90 and 135 is indeed 45.