Problem 14 - 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. $$2 \odot 7$$

Step-by-Step Solution

Answer

The greatest common divisor of 2 and 7 is 1. So, \(2.7 = \operatorname{gcd}(2, 7) = 1 \).

Step 1: Identify the Expression to Factor

We need to factor the expression: \(2 \odot 7\).

Step 2: Look for Common Factors and Patterns

We examine the expression to find common factors, patterns (like difference of squares, perfect square trinomials, grouping), or use appropriate factoring techniques.

Step 3: Write the Factored Form

The greatest common divisor of 2 and 7 is 1. So, \(2.7 = \operatorname{gcd}(2, 7) = 1 \).

Key Concepts

Greatest Common Divisor

The greatest common divisor, also known as gcd, is a fundamental concept in number theory. It represents the largest number that divides two or more integers without leaving a remainder. To determine the gcd of two numbers, one method is to list the divisors of each number and then identify the largest divisor common to both.

When dealing with prime numbers, the gcd can be quite simple. Since prime numbers like 2 and 7 have no other divisors besides 1 and themselves, their gcd will be 1 if they differ. For composite numbers, you can use strategies like prime factorization to find the gcd.

• Write down the prime factors of each number.
• Identify the common factors.
• Select the largest of these common factors as the gcd.

Understanding the gcd is crucial because it is used in simplifying fractions, determining to simplify ratios, and even in more complex mathematical structures like rings and fields. In our exercise, since 2 and 7 are prime, the gcd is simply 1, as there are no other common divisors.

Least Common Multiple

The least common multiple, or lcm, is another essential concept that complements the gcd. It refers to the smallest non-zero common multiple of two or more numbers. For many students, finding the lcm can be a bit tricky, but it is essential for problems involving fractions and discrete mathematics topics.

To find the lcm of two numbers, you can apply the following method:

• List the multiples of each number.
• Identify the smallest multiple that appears in both lists.
• This smallest multiple is the lcm.

The lcm is especially useful when you want to add fractions with different denominators. It helps find a common denominator, which is the lcm of the original denominators.

Mathematically, the relationship between the gcd and lcm of two numbers can be expressed as:\[\text{gcd}(a, b) \times \text{lcm}(a, b) = a \times b\]Knowing one can help you find the other when dealing with two numbers. This relationship is frequently used in problems requiring both gcd and lcm.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with objects that can take on distinct, separate values. It includes topics like graphs, logic, set theory, and number theory, which are fundamental in computer science and combinatorics.

When looking at problems involving the gcd and lcm, these topics often fall under the domain of discrete mathematics. These concepts don't deal with continuous values; instead, they're concerned with distinct entities, such as integers or sets.

• Discrete mathematics involves algorithms, crucial for computing gcd and lcm efficiently.
• It's applied in coding theory, cryptography, and network analysis.
• Boolean algebra, an integral part of discrete mathematics, manipulates truth values and logical statements.

In the provided exercise, the set of factors of 70 forms a boolean algebra, showing the connection between set theory and discrete mathematics. Boolean algebra is particularly useful in the design and simplification of circuits and logic gates, which are pivotal in modern computing.