What Is The Greatest Common Factor Of 42 And 90

What is the Greatest Common Factor of 42 and 90? A Deep Dive into Number Theory

Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers might seem like a simple arithmetic task. However, understanding the underlying principles and exploring different methods for calculating the GCF provides a fascinating glimpse into number theory and its practical applications. This article delves deep into finding the GCF of 42 and 90, explaining various methods and highlighting the broader mathematical concepts involved.

Understanding the Greatest Common Factor (GCF)

The greatest common factor (GCF) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. It's a fundamental concept in number theory with applications in various fields, including cryptography, computer science, and music theory. For instance, understanding GCFs is crucial for simplifying fractions to their lowest terms.

In our case, we want to find the GCF of 42 and 90. This means we are looking for the largest number that perfectly divides both 42 and 90.

Method 1: Prime Factorization

This is arguably the most straightforward and conceptually clear method. It involves finding the prime factorization of each number and then identifying the common prime factors raised to the lowest power.

Step 1: Prime Factorization of 42

42 can be broken down into its prime factors as follows:

42 = 2 × 21 = 2 × 3 × 7

Therefore, the prime factorization of 42 is 2 × 3 × 7.

Step 2: Prime Factorization of 90

90 can be factorized as:

90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5 = 2 × 3² × 5

Thus, the prime factorization of 90 is 2 × 3² × 5.

Step 3: Identifying Common Factors

Now, we compare the prime factorizations of 42 and 90:

42 = 2 × 3 × 7 90 = 2 × 3² × 5

The common prime factors are 2 and 3. The lowest power of 2 that appears in both factorizations is 2¹, and the lowest power of 3 is 3¹.

Step 4: Calculating the GCF

To find the GCF, we multiply the common prime factors raised to their lowest powers:

GCF(42, 90) = 2¹ × 3¹ = 2 × 3 = 6

Therefore, the greatest common factor of 42 and 90 is 6.

Method 2: Listing Factors

This method is simpler for smaller numbers but becomes less efficient as numbers get larger. It involves listing all the factors of each number and then identifying the largest common factor.

Step 1: Factors of 42

The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

Step 2: Factors of 90

The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

Step 3: Identifying Common Factors

Comparing the two lists, the common factors are 1, 2, 3, and 6.

Step 4: Determining the GCF

The largest common factor is 6. Therefore, the GCF(42, 90) = 6.

Method 3: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF of two integers, especially when dealing with 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.

Step 1: Applying the Algorithm

We start with the numbers 42 and 90. Since 90 > 42, we replace 90 with 90 - 42 = 48. Now we have the pair 42 and 48.

Next, we repeat the process: 48 - 42 = 6. Now we have 42 and 6.

Again: 42 - 6 = 36. Now we have 36 and 6.

And again: 36 - 6 = 30. Now we have 30 and 6.

And again: 30 - 6 = 24. Now we have 24 and 6.

And again: 24 - 6 = 18. Now we have 18 and 6.

And again: 18 - 6 = 12. Now we have 12 and 6.

And again: 12 - 6 = 6. Now we have 6 and 6.

Since both numbers are now equal, the GCF is 6.

Step 2: A More Efficient Euclidean Algorithm Formulation

The Euclidean algorithm can be expressed more efficiently using modulo operation (%):

1. Divide the larger number by the smaller number and find the remainder.
2. Replace the larger number with the smaller number and the smaller number with the remainder.
3. Repeat steps 1 and 2 until the remainder is 0. The last non-zero remainder is the GCF.

Let's apply this to 42 and 90:

• 90 % 42 = 6
• 42 % 6 = 0

The last non-zero remainder is 6, therefore GCF(42, 90) = 6.

Applications of the GCF

The GCF has numerous practical applications across various disciplines:

• Simplifying Fractions: The GCF is used to simplify fractions to their lowest terms. For example, the fraction 42/90 can be simplified to 7/15 by dividing both the numerator and denominator by their GCF, which is 6.

• Solving Word Problems: Many word problems in mathematics involve finding the GCF. For instance, problems related to dividing objects into equal groups or finding the largest possible size of square tiles to cover a rectangular floor utilize the GCF concept.

• Cryptography: The GCF plays a crucial role in various cryptographic algorithms, such as the RSA algorithm, which relies on the difficulty of finding the GCF of two very large numbers.

• Computer Science: GCF calculations are used in various computer algorithms, particularly in tasks related to data compression and optimization.

• Music Theory: Understanding GCFs is important in music theory when dealing with musical intervals and creating harmonies.

Conclusion: The Power of Number Theory

Finding the greatest common factor of 42 and 90, while seemingly a simple arithmetic problem, reveals the beauty and power of number theory. The different methods presented – prime factorization, listing factors, and the Euclidean algorithm – each offer unique insights into the mathematical principles involved. Understanding these methods not only provides a solid foundation in number theory but also equips you with valuable tools applicable to various fields and problem-solving situations. The GCF, a seemingly small concept, holds significant weight in the larger mathematical landscape, demonstrating the interconnectedness and utility of even seemingly elementary mathematical concepts. It underscores the importance of exploring seemingly simple mathematical concepts to their full depth, unveiling their rich applications and broader significance.