Problem 19 - 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. $$10 \oplus 10$$

Step-by-Step Solution

Answer

The short answer to the given expression \(10+10\) using the operations provided is: \(10+10 = \operatorname{lcm}(10, 10) = 10\).

Step 1: Find the prime factors of each number

Find the prime factors of each number: \(10 = 2 \cdot 5\)

Step 2: Calculate the LCM

Since the prime factors of both numbers are the same, we will simply use them to find the LCM. The LCM of 10 and 10 is: \(10 = 2 \cdot 5\) Now we have: \[10+10 = \operatorname{lcm}(10, 10) = 10\]

Key Concepts

Least Common Multiple

The concept of the Least Common Multiple (LCM) is an essential part of understanding various aspects of number theory and arithmetic operations. It helps us find the smallest multiple that two or more numbers share.
To calculate the LCM of two numbers, you need to check their prime factors. For example, if you are finding the LCM of 10 and 10, both numbers have the same prime factors:

• 2 and 5 for 10

The LCM of any two numbers multiplied by themselves, as in this case, remains the same, as they share identical prime factors. Hence, the least common multiple of 10 and 10 is simply 10.
Think of LCM as finding a common stage where both numbers can appear without any conflict. It's a fundamental tool in algebra, helping simplify problems involving ratios, synchronization, and combined frequency of events.

Greatest Common Divisor

The Greatest Common Divisor (GCD), also known as the greatest common factor, is the largest number that divides two or more numbers without leaving a remainder. Unlike the LCM, instead of finding a common multiple, the GCD helps us find a common factor.
When calculating the GCD of the same number, such as 10 and 10, we again look at their prime factors, which for 10 are:

• 2 and 5

Since both numbers are the same, the greatest common divisor is simply the number itself, which in this case is 10.
Understanding GCD is crucial for simplifying fractions, optimizing algorithms, and solving problems in modular arithmetic. Think of GCD as the building block that shows the maximum shared part between numbers.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For instance, in the prime factorization of 10, we find:

• 2, which is the smallest prime number
• 5, another prime number

Thus, 10 can be expressed as the product of 2 and 5. Prime factorization is extremely valuable since it provides insights into the structure of numbers, allowing us to solve problems related to divisibility, LCM, and GCD. It can also assist in understanding more complex mathematical concepts like cryptography and number field theory. Think of it like the atoms in a molecule, providing the fundamental composition of any number.