Problem 8 - A Book of Abstract Algebra

Question

If \(a \mid b\) and \(c \mid d\), then \(a c \mid b d\).

Step-by-Step Solution

Answer

If \(a\mid b\) and \(c\mid d\), then \(ac\mid bd\) is true since \(bd = ac \cdot (k \cdot m)\).

Step 1: Understand the Given Information

The problem states that if \(a\) divides \(b\) and \(c\) divides \(d\), then \(ac\) must divide \(bd\). This means there exist integers \(k\) and \(m\) such that \(b = a \cdot k\) and \(d = c \cdot m\).

Step 2: Express the Product bd in Terms of a, c, k, and m

Using the expressions from Step 1, substitute \(b\) and \(d\) in the product \(bd\): \(bd = (a \cdot k)(c \cdot m) = a \cdot c \cdot (k \cdot m)\).

Step 3: Demonstrate that ac Divides bd

From Step 2, we have expressed \(bd\) as \(bd = ac \cdot (k \cdot m)\). Since \(k \cdot m\) is an integer, \(ac\) divides \(bd\) by definition of divisibility.

Key Concepts

Divisibility

In abstract algebra, divisibility plays a crucial role in understanding how integers interact with each other. When we say a number "divides" another, denoted as \(a \mid b\), it means there exists an integer \(k\) such that \(b = a \times k\). This relationship provides a framework for numerous mathematical concepts.

Divisibility ensures a perfect division without any remainder. When solving problems like the one given, it's important to establish the presence of integers \(k\) and \(m\) that highlight how one number can be expressed as a multiple of another. In essence, divisibility is about finding consistency and predictability in numbers, which can simplify complex problems drastically.

Integers

Integers are the whole numbers that can be positive, negative, or zero. They form the basic building blocks of algebra. Understanding integers is fundamental when exploring concepts like divisibility and multiplication.

These numbers are used in divisibility rules to determine how they can divide each other without leaving a remainder. In the problem discussed, integers such as \(k\) and \(m\) are used to represent the factors that help demonstrate how one number can be expressed through multiplication of its divisor by these integers. Exploring algebra with integers enhances problem-solving by providing clear pathways through otherwise complex mathematical challenges.

Multiplication

Multiplication is one of the basic operations in mathematics. It involves finding the product of two numbers, denoted as \(a \times b\). In algebra, especially when discussing divisibility and integers, multiplication helps express relationships between numbers.

For example, determining if \(ac\) divides \(bd\) implies forming a product where each number is multiplied by a respective integer factor. In the example given, the equation \(bd = ac \cdot (k \cdot m)\) uses multiplication to simplify and confirm the divisibility condition. This operation consolidates the inter-relationships between variables, making complex equations solvable.

Problem-Solving

Problem-solving is the process of finding solutions to complex or challenging issues. In algebra, it often involves breaking down abstract problems into smaller, more manageable parts.

The exercise given shows a structured approach to problem-solving. By understanding what is given, expressing important parts in known variables, and verifying results through definitions (like divisibility), the problem is solved step-by-step. This approach helps avoid overwhelming confusion and builds confidence in handling more complicated algebraic expressions. Recognizing the roles of divisibility, integers, and multiplication enhances our ability to approach problems logically and without error.