Problem 49 - Discrete Mathematics with Applications

Question

Disprove each statement. If \(\operatorname{gcd}\\{a, b\\}=1\) and \(\operatorname{gcd}\\{b, c\\}=1,\) then \(\operatorname{gcd}\\{a, c\\}=1,\) where \(a, b,\) and \(c\) are positive integers.

Step-by-Step Solution

Answer

We can disprove the statement with an example: Let \(a = 2\), \(b = 3\), and \(c = 6\). Then, \(\operatorname{gcd}(2, 3) = 1\) and \(\operatorname{gcd}(3, 6) = 3\), but \(\operatorname{gcd}(2, 6) = 2 \neq 1\). This shows that the statement is not always true.

Step 1: Find an Example

We can choose a, b, and c as follows: a = 2, b = 3, and c = 6.

Step 2: Compute gcd(a, b)

We need to find gcd(2, 3), the greatest common divisor of 2 and 3. Since 2 and 3 are prime numbers and have no common factors other than 1, gcd(2, 3) = 1.

Step 3: Compute gcd(b, c)

We need to find gcd(3, 6), the greatest common divisor of 3 and 6. Since 3 is a factor of 6, gcd(3, 6) = 3.

Step 4: Compute gcd(a, c)

We need to find gcd(2, 6), the greatest common divisor of 2 and 6. Since 2 is a factor of 6, gcd(2, 6) = 2.

Step 5: Check the Conditions

We indeed have gcd(a, b) = 1 and gcd(b, c) = 1, but gcd(a, c) = 2 ≠ 1. So, the given statement is not always true.

Key Concepts

Number Theory

Number theory is a fascinating branch of mathematics focused on the properties and relationships of numbers, particularly integers. It's the study of the set of positive whole numbers which are usually called the set of natural numbers. Number theory is often considered the purest form of mathematical inquiry, delving into the world of numbers without an immediate application. Yet, it has given rise to some of the most profound and beautiful ideas in mathematics, influencing fields as diverse as cryptology and computer science.

When we discuss the greatest common divisor (gcd) within number theory, we are essentially examining the largest positive integer that divides two or more integers without leaving a remainder. Finding the gcd is a fundamental concept and is a stepping stone to more advanced topics in number theory. The properties of the greatest common divisor play an essential role in various theorem proofs, algorithm developments, and encryption methods like the RSA algorithm in cryptography.

In our textbook example on gcd, we take into consideration various positive integers and use steps to calculate the gcd. Applying the basic principles of number theory allows us to deeply understand the relationships between different numbers, such as why the gcd of prime numbers that are different is always 1.

Prime Numbers

Prime numbers are the building blocks of number theory. They are special because a prime number is only divisible by 1 and itself. The intriguing aspect of prime numbers is that every integer greater than 1 is either a prime number or can be uniquely factored into prime numbers, which is known as the Fundamental Theorem of Arithmetic.

In the given step-by-step solution, the numbers 2 and 3 are prime numbers, and thus, their greatest common divisor (gcd) is 1, since they have no other common factors. This characteristic is foundational when exploring number theory and its applications. Understanding how prime numbers work is key to dissecting more complicated number theory problems, determining least common multiples, or even breaking down security algorithms in digital security.

Furthermore, the uniqueness of prime factorization is immensely useful because it gives us a profound insight into the fabric of numerical systems and the way numbers interact.

Counterexample Method

The counterexample method is a crucial proof technique in mathematics, particularly useful in disproving statements or conjectures. Instead of proving that a statement is true, we provide a single instance, or 'counterexample,' where the statement does not hold, and this disproves the universal claim being made.

In the context of our textbook exercise, the counterexample method is elegantly deployed. By choosing specific values for a, b, and c - where a = 2, b = 3, and c = 6 - we find that while gcd(2, 3) = 1 and gcd(3, 6) = 1 are true, the statement gcd(2, 6) = 1 is false. Thus, the counterexample successfully demonstrates that the initial assertion about gcd values was incorrect. This method highlights the importance of critical thinking and careful selection of cases in mathematical problem-solving. The ability to disprove by counterexample is a powerful tool that sharpens critical analysis and reinforces the understanding that mathematical proofs are not always intuitive.