Problem 1 - A Book of Abstract Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Divisibility

First, recall what it means for one number to divide another. If \(a \mid b\), this means that there exists an integer \(k\) such that \(b = a \times k\). Similarly, if \(b \mid c\), there exists an integer \(m\) such that \(c = b \times m\).

Step 2: Substitute and Expand

Substitute the expression for \(b\) from the first divisibility condition into the second. Since \(b = a \times k\), replace \(b\) in \(c = b \times m\) to get \(c = (a \times k) \times m\).

Step 3: Rearrange the Expression

Rearrange the expression obtained in Step 2: \(c = a \times (k \times m)\). Notice that \(k \times m\) is an integer (since the product of two integers is an integer).

Step 4: Conclude Divisibility

Since \(c = a \times (k \times m)\) and \(k \times m\) is an integer, we conclude that \(a \mid c\). Therefore, \(a\) divides \(c\).

Key Concepts

Integer Multiplication

In mathematical terms, multiplication is a fundamental arithmetic operation that involves combining numbers to form a product. When we talk about integer multiplication, it simply means multiplying whole numbers either positive, negative, or zero. Let's break down what this looks like in the context of divisibility.
When you multiply integers, the result or product of those integers is also an integer. This property is crucial when dealing with problems related to divisibility, as seen in the exercise where we have conditions like \(a \mid b\) and \(b \mid c\).

• If \(b = a \times k\) for some integer \(k\), then \(b\) is a multiple of \(a\).
• Furthermore, if \(c = b \times m\), then \(c\) is also related to \(b\) by the integer \(m\).

The multiplication here creates a chain - first between \(a\) and \(b\), then between \(b\) and \(c\). This chaining is what allows us to expand and manipulate expressions in proofs dealing with divisibility.

Mathematical Proofs

A mathematical proof is a logical argument that establishes the truth of a given statement using accepted mathematical principles. Proofs are essential in mathematics as they provide a foundation for the validity of statements or theorems. The exercise above illustrates a classic example of using proofs to show divisibility.
In the step-by-step solution:

• We start by defining the terms: knowing \(a \mid b\) implies \(b = a \times k\), and \(b \mid c\) implies \(c = b \times m\).
• These definitions are then logically combined: Substitute \(b = a \times k\) into \(c = b \times m\) to get \(c = (a \times k) \times m\).
• This leads to the rearrangement \(c = a \times (k \times m)\).

Such manipulations are meant to clearly establish the transitivity of divisibility, ensuring that each step is logically sound, thereby proving \(a \mid c\). These kinds of proofs help to strengthen mathematical understanding by tracing through the logical steps.

Transitive Property of Divisibility

The transitive property is an important concept in math, meaning that if some property is true between one pair of elements and then between a second pair, it must be true between the first and last elements as well. For divisibility, this means that if \(a \mid b\) and \(b \mid c\), then it must follow that \(a \mid c\).

• This transitive nature ensures a continuous relation through integers, linking \(a\), \(b\), and \(c\) in a way that establishes consistency across mathematical operations.
• In our example, once substituting and rearranging the expression to \(c = a \times (k \times m)\), we see that since \(k \times m\) is an integer, \(a\) divides \(c\).
• This shows that divisibility works similarly to a bridge: if \(a\) can cross to \(b\) and \(b\) can reach \(c\), then \(a\) must reach \(c\).

Understanding this property helps simplify complex problems by allowing us to infer results across a chain of divisibility relationships without having to calculate each step explicitly.