Greatest Common Factor Of 42 And 90

Finding the Greatest Common Factor (GCF) of 42 and 90: 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 ranging from simplifying fractions to solving algebraic equations. This comprehensive guide will delve into multiple methods for determining the GCF of 42 and 90, explaining each step in detail and exploring the underlying mathematical principles. We'll also examine the broader context of GCFs and their importance in various mathematical areas.

Understanding the Greatest Common Factor (GCF)

Before we embark on calculating the GCF of 42 and 90, let's solidify our understanding of the concept. The greatest common factor 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 goes into both numbers evenly.

For example, let's consider the numbers 12 and 18. 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 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 simplest method, although potentially time-consuming for larger numbers, involves listing all the factors of each number and identifying the largest common factor.

Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Comparing the two lists, we identify the common factors: 1, 2, 3, and 6. The greatest of these is 6. Therefore, the GCF of 42 and 90 is 6. This method is straightforward but becomes less practical when dealing with larger numbers with many factors.

Method 2: Prime Factorization

This method utilizes the prime factorization of each number. Prime factorization involves expressing a number as a product of its prime factors – numbers divisible only by 1 and themselves.

Prime Factorization of 42:

42 = 2 × 21 = 2 × 3 × 7

Prime Factorization of 90:

90 = 2 × 45 = 2 × 5 × 9 = 2 × 3² × 5

Now, we identify the common prime factors and their lowest powers present in both factorizations. Both 42 and 90 share a single factor of 2 and a single factor of 3.

Therefore, the GCF is 2 × 3 = 6.

This method is more efficient than listing factors, especially when dealing with larger numbers, as it systematically breaks down the numbers into their fundamental prime components.

Method 3: Euclidean Algorithm

The Euclidean Algorithm provides a highly efficient method for finding the GCF, especially useful 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 42 and 90:

90 = 2 × 42 + 6 (We divide 90 by 42, the quotient is 2, and the remainder is 6)
42 = 7 × 6 + 0 (We divide 42 by the remainder 6, the quotient is 7, and the remainder is 0)

When the remainder becomes 0, the GCF is the last non-zero remainder, which is 6. The Euclidean Algorithm provides a systematic and efficient approach, even for significantly larger numbers, avoiding the need for extensive factorization or factor listing.

Applications of GCF

The GCF has numerous applications across various mathematical fields and real-world scenarios:

1. Simplifying Fractions:

The GCF is crucial for simplifying fractions to their lowest terms. For instance, the fraction 42/90 can be simplified by dividing both the numerator and denominator by their GCF, which is 6:

42/90 = (42 ÷ 6) / (90 ÷ 6) = 7/15

This simplifies the fraction to its simplest form.

2. Solving Algebraic Equations:

GCF plays a role in factoring algebraic expressions. When factoring polynomials, finding the GCF of the terms allows us to simplify the expression and potentially solve equations more easily.

3. Real-World Applications:

GCF finds applications in various practical situations:

Dividing objects: Imagine you have 42 apples and 90 oranges, and you want to divide them into identical groups with the maximum number of items in each group. The GCF (6) determines the maximum number of identical groups you can create. Each group would have 7 apples and 15 oranges.
Geometry: GCF can be used in geometry problems involving finding the dimensions of squares or rectangles with a given area.
Measurement: In situations requiring precise measurements, the GCF helps in finding common units or subdivisions.

Understanding Prime Numbers and their Role in GCF

Prime numbers, the building blocks of all integers, are essential to understanding GCF. Prime numbers are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...). The prime factorization method for finding the GCF relies heavily on identifying the prime factors of each number involved. The fundamental theorem of arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime numbers, disregarding the order of the factors. This uniqueness ensures the consistency and reliability of the prime factorization method for GCF calculation.

Exploring Least Common Multiple (LCM) and its Relationship with GCF

The least common multiple (LCM) is another important concept closely related to the GCF. The LCM of two or more integers is the smallest positive integer that is divisible by all the given integers. For example, the LCM of 42 and 90 is 630. The GCF and LCM are connected through the following relationship:

LCM(a, b) × GCF(a, b) = a × b

Where 'a' and 'b' are the two integers. In the case of 42 and 90:

LCM(42, 90) × GCF(42, 90) = 42 × 90

630 × 6 = 3780

This relationship provides a useful check on the accuracy of GCF and LCM calculations.

Conclusion

Finding the greatest common factor is a fundamental mathematical skill with widespread applications. While the listing factors method provides a simple approach for smaller numbers, the prime factorization and Euclidean Algorithm methods offer more efficient solutions for larger numbers. Understanding the GCF is crucial for simplifying fractions, solving equations, and tackling various practical problems. The relationship between the GCF and LCM further enriches our understanding of number theory and its applications. Mastering these concepts forms a solid foundation for more advanced mathematical studies.