Common Factors For 84 And 105

Unraveling the Common Factors of 84 and 105: A Deep Dive into Number Theory

Finding the common factors of two numbers might seem like a simple arithmetic exercise, but it's a fundamental concept in number theory with broader applications in mathematics and computer science. This article explores the common factors of 84 and 105, demonstrating various methods to identify them, and delving into the underlying principles. We'll cover prime factorization, greatest common divisors (GCD), and least common multiples (LCM), ultimately offering a comprehensive understanding of this seemingly straightforward topic.

Understanding Factors and Common Factors

Before we delve into the specifics of 84 and 105, let's establish a clear understanding of the terminology.

Factors: A factor of a number is any integer that divides the number evenly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Common Factors: Common factors are numbers that are factors of two or more different numbers. For instance, the common factors of 12 and 18 are 1, 2, 3, and 6.

Method 1: Prime Factorization

Prime factorization is a powerful technique for finding the common factors of any two numbers. It involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves. Let's apply this method to 84 and 105:

Prime Factorization of 84

We can systematically break down 84 into its prime factors:

84 = 2 x 42
42 = 2 x 21
21 = 3 x 7

Therefore, the prime factorization of 84 is 2² x 3 x 7.

Prime Factorization of 105

Similarly, let's find the prime factorization of 105:

105 = 3 x 35
35 = 5 x 7

Therefore, the prime factorization of 105 is 3 x 5 x 7.

Identifying Common Factors through Prime Factorization

By comparing the prime factorizations of 84 (2² x 3 x 7) and 105 (3 x 5 x 7), we can easily identify the common prime factors: 3 and 7.

To find all the common factors, we consider all possible combinations of these common prime factors:

1: (The product of no common prime factors)
3: (The product of the common prime factor 3)
7: (The product of the common prime factor 7)
21: (The product of the common prime factors 3 and 7)

Therefore, the common factors of 84 and 105 are 1, 3, 7, and 21.

Method 2: Listing Factors

A more straightforward, though potentially time-consuming for larger numbers, method involves listing all the factors of each number and then identifying the common ones.

Factors of 84:

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Factors of 105:

1, 3, 5, 7, 15, 21, 35, 105

Identifying Common Factors from Lists

Comparing the two lists, we again find the common factors to be 1, 3, 7, and 21.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the largest of the common factors. In the case of 84 and 105, the GCD is 21. The GCD is crucial in various mathematical applications, including simplifying fractions and solving Diophantine equations.

Calculating GCD using the Euclidean Algorithm

The Euclidean algorithm provides an efficient method for finding the GCD of two numbers. It's based on the principle that the GCD of two numbers remains unchanged if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal, representing the GCD.

Let's apply the Euclidean algorithm to 84 and 105:

105 = 1 x 84 + 21
84 = 4 x 21 + 0

Since the remainder is 0, the GCD is the last non-zero remainder, which is 21.

Least Common Multiple (LCM)

While we've focused on common factors, the least common multiple (LCM) is another important concept related to the factors of two numbers. The LCM is the smallest positive number that is a multiple of both numbers.

Calculating LCM using Prime Factorization

We can calculate the LCM using the prime factorizations we found earlier:

Prime factorization of 84: 2² x 3 x 7
Prime factorization of 105: 3 x 5 x 7

To find the LCM, we take the highest power of each prime factor present in either factorization:

LCM(84, 105) = 2² x 3 x 5 x 7 = 420

Relationship between GCD and LCM

There's an important relationship between the GCD and LCM of two numbers (a and b):

LCM(a, b) x GCD(a, b) = a x b

Let's verify this relationship for 84 and 105:

LCM(84, 105) x GCD(84, 105) = 420 x 21 = 8820

84 x 105 = 8820

The equation holds true, confirming the relationship between GCD and LCM.

Applications of Common Factors and GCD

Understanding common factors and the GCD has numerous applications across various fields:

Simplifying Fractions: Finding the GCD allows us to simplify fractions to their lowest terms. For example, the fraction 84/105 can be simplified to 4/5 by dividing both the numerator and denominator by their GCD (21).
Cryptography: GCD plays a crucial role in algorithms used in public-key cryptography, such as the RSA algorithm.
Scheduling Problems: Finding the LCM helps solve scheduling problems where events need to occur at regular intervals. For example, if two events occur every 84 and 105 days, the LCM (420) determines when they will coincide again.
Computer Science: GCD and LCM are used in algorithms related to data structures and computer graphics.

Conclusion

Finding the common factors of 84 and 105, while seemingly a simple task, provides a gateway to understanding fundamental concepts in number theory. Through prime factorization, listing factors, and applying the Euclidean algorithm, we've identified the common factors (1, 3, 7, 21), the GCD (21), and the LCM (420). Understanding these concepts and their interrelationships is vital for numerous applications in mathematics and computer science. The exploration goes beyond simple arithmetic; it's a journey into the elegant structure of numbers and their properties. The methods discussed offer both theoretical understanding and practical tools for solving related problems. Remember, mastery in mathematics often arises from a deep understanding of seemingly simple concepts.