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Chapter 5: Problem 28

Let \(m, n,\) and \(d\) be integers. Show that if \(d \mid m,\) then \(d \mid m n\).

Short Answer

Expert verified
If \( d \) divides \( m \), denoted as \( d \mid m \), then \( d \) also divides \( m \times n \), or \(d \mid m \times n\). This is true for all integers \( m, n, \) and \( d \).

Step by step solution

01

Understand the Notation

The notation \(d \mid m\) means \(d\) divides \(m\). If \(d\) divides \(m\), then there exists an integer \(k\) such that \(d \times k = m\). Our goal is to show that under these conditions, \(d\) also divides \(m \times n\), or \(d \mid m \times n\).
02

Consider \(m\) and \(n \)

As per the given condition, \(m = d \times k\) for some integer \(k\). Now consider the product \(m \times n\). \(m \times n = (d \times k) \times n = d \times (k \times n)\). Let's take \(k \times n\) as some integer \(p\).
03

Establish Division

From the earlier step, \(m \times n = d \times p\), where \(p\) is an integer (since the product of two integers is also an integer). Hence, we established that \(d \mid m \times n \), meaning, \(d\) divides \(m \times n \). This confirms the given statement is true and concludes our step-by-step solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integer Division
Integer division is a fundamental concept in mathematics, where one integer, known as the divisor, divides another, known as the dividend. This occurs when there's an integer quotient with no remainder. In our exercise, we are analyzing divisibility, specifically looking at the notation \( d \mid m \), which signifies that an integer \( d \) divides another integer \( m \).
  • For example, if \( d = 3 \) and \( m = 12 \), then 3 perfectly divides 12, such that the quotient is 4, hence proving divisibility.
  • We express this as \( d \mid m \), meaning there exists an integer \( k \) such that \( m = d \times k \).
Understanding integer division helps in simplifying arithmetic operations and allows us to prove other mathematical properties, as shown in the exercise where we extend this concept to multiple terms.
Discrete Mathematics
Discrete mathematics deals with countable, distinct elements and is essential for developing logical and algorithmic thinking. It covers a broad range of topics including numbers, logic, graphs, and sets, but within our context, we focus on divisibility.
  • In our problem, we are handling integers, a crucial part of discrete math. The concept of divisibility is integral to number theory, a significant branch of discrete mathematics.
  • We often use simple yet powerful properties of numbers, such as divisibility rules, to analyze and provide insights into more complex problems.
Through this lens, discrete mathematics provides the framework to analyze the divisibility properties of integers, linking fundamental arithmetic operations to sophisticated mathematical proofs.
Proof Techniques
Proof techniques in mathematics provide a systematic means to verify whether mathematical statements hold true. For the problem at hand, we employ the technique of direct proof, a method commonly used in discrete mathematics.
  • A direct proof starts with known truths or assumptions and uses logical steps to arrive at a conclusion. In our exercise, we assume \( d \mid m \) and directly use the definition to show \( d \mid mn \).
  • By expressing \( m \) as \( d \times k \), we only need to illustrate that multiplying it by another integer \( n \) still keeps it divisible by \( d \).
Understanding proof techniques not only assists in solving particular problems but also enhances logical reasoning and problem-solving skills applicable across various domains of mathematics.

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Most popular questions from this chapter

Show that \(\operatorname{gcd}(n, \phi)=1,\) and find the inverse s of \(n\) modulo \(\phi\) satisfying \(0

Use the following definition: \(A\) subset \(\left\\{a_{1}, \ldots, a_{n}\right\\}\) of \(\mathbf{Z}^{+}\) is \(a^{*}\) -set of size \(n\) if \(\left(a_{i}-a_{j}\right) \mid a_{i}\) for all \(i\) and \(j,\) where \(i \neq j, 1 \leq i \leq n,\) and \(1 \leq j \leq n .\) These exercises are due to Martin Gilchrist. Prove that for all \(n \geq 2,\) there exists a \(*\) -set of size \(n .\) Hint: Use induction on \(n .\) For the Basis Step, consider the set \\{1,2\\} For the Inductive Step, let \(b_{0}=\prod_{k=1}^{n} a_{k}\) and \(b_{i}=b_{0}+a_{i}\) for \(1 \leq i \leq n\).

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