Problem 16 - Discrete Mathematics with Applications

Question

The set \(D_{70}=\\{1,2,5,7,10,14,35,70\\}\) of positive factors of 70 is a boolean algebra under the operations \(\oplus, \odot,\) and ' defined by \(x \oplus y=\operatorname{lcm}\\{x, y\\}\) \(x \odot y=\operatorname{gcd}\\{x, y\\},\) and \(x^{\prime}=70 / x .\) Compute each. $$5^{\prime} \odot 7^{\prime}$$

Step-by-Step Solution

Answer

The short answer for the given exercise is: \(5^{\prime} . 7^{\prime} = 2\).

Step 1: Compute 5' and 7'

First, we need to calculate the result of applying the prime operation on the given numbers. According to the definition of the prime operation, we have: \(5^{\prime} = \frac{70}{5} = 14\) \(7^{\prime} = \frac{70}{7} = 10\)

Step 2: Calculate the GCD of 5' and 7'

Now that we have calculated 5' and 7', we need to find their GCD, which is represented by the dot operation. The greatest common divisor (GCD) refers to the largest positive integer that divides two numbers without leaving a remainder. In this case, we want to find the greatest common divisor of 14 and 10. We can use the Euclidean algorithm for this purpose: 1. Divide the larger number (14) by the smaller number (10). 2. Then divide the previous divisor (10) by the remainder from the first division (4). 3. Continue this process until the remainder is 0. Applying the Euclidean algorithm: \(14 = 1 \times 10 + 4\) (remainder: 4) \(10 = 2 \times 4 + 2\) (remainder: 2) \(4 = 2 \times 2 + 0\) (remainder: 0) The last non-zero remainder is 2, which is the GCD of 14 and 10.

Step 3: The result

Now that we have computed the GCD of 5' and 7', we have the result of applying the dot operation on these two numbers: \(5^{\prime} . 7^{\prime} = GCD(14, 10) = 2\) So, the answer to the given exercise is: \(5^{\prime} . 7^{\prime} = 2\).

Key Concepts

Greatest Common Divisor

Understanding the Greatest Common Divisor (GCD), also known as the greatest common factor or highest common divisor, is crucial in various areas of mathematics and its applications, including simplifying fractions and solving Diophantine equations.

The GCD of two numbers is the largest positive integer that evenly divides both numbers without leaving a remainder. For example, consider the numbers 8 and 12; the divisors of 8 are \(1, 2, 4, 8\), and the divisors of 12 are \(1, 2, 3, 4, 6, 12\). The common divisors are \(1, 2, 4\), with 4 being the greatest.

When working with more complex numbers or when the numbers are not easily factorable, various algorithms, including the Euclidean algorithm, can prove to be powerful tools to calculate the GCD.

Least Common Multiple

The Least Common Multiple (LCM) is the smallest non-zero common multiple of two or more numbers. It's particularly useful for finding common denominators in addition and subtraction of fractions, and for solving problems involving time, such as when two events will next occur at the same time.

Going back to our example with the numbers 8 and 12, the multiples of 8 are \(8, 16, 24, 32, \) and so on, while the multiples of 12 are \(12, 24, 36, \) and so on. The common multiples are numbers that appear in both lists - the first of which is 24. Therefore, the LCM of 8 and 12 is 24.

Calculating the LCM can be done through listing out multiples, using prime factorization, or applying the GCD with the formula \(LCM(a, b) = \frac{|a \times b|}{GCD(a, b)}\).

Euclidean Algorithm

The Euclidean algorithm is a method for finding the GCD of two integers and is one of the oldest algorithms known, still essential today for its simplicity and efficiency.

To employ this algorithm, start by dividing the larger number by the smaller and take the remainder. Then, divide the smaller number by this remainder. Repeat the process: divide the last divisor by the last remainder. When you reach a remainder of zero, the last non-zero remainder is the GCD.

Let's apply this algorithm to the numbers 14 and 10 from our original problem:

• \( 14 = 1 \times 10 + 4 \) (first remainder is 4)
• \( 10 = 2 \times 4 + 2 \) (second remainder is 2)
• \( 4 = 2 \times 2 + 0 \) (third remainder is 0)

The process stops here, as the last remainder is zero, and the GCD is the last non-zero remainder, which is 2. This elegant and straightforward algorithm was employed to solve our exercise, showcasing not only a deep historical mathematical concept but also a practical problem-solving tool.