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Chapter 3: Problem 41

The difference of any two odd integers is odd.

Short Answer

Expert verified
The given statement is false. The difference of any two odd integers is not odd, but even. This can be demonstrated using the general form of odd integers: \(a = 2n_1 + 1\) and \(b = 2n_2 + 1\). Subtracting 'b' from 'a' and simplifying, we obtain \(c = 2(n_1 - n_2)\). Since 'c' is the product of 2 and an integer, it must be even.

Step by step solution

01

Understand odd integers

An integer is odd if it is not divisible by 2, i.e., it has a remainder of 1 when divided by 2. The general form of an odd integer can be written as 2n + 1, where n is any integer.
02

Write the general form of two odd integers

Let's consider two odd integers 'a' and 'b'. Using the general form of odd integers mentioned in Step 1, we can write odd integer 'a' as \(a = 2n_1 + 1\) and 'b' as \(b = 2n_2 + 1\), where \(n_1\) and \(n_2\) are integers.
03

Calculate the difference of the two odd integers

Subtract integer 'b' from 'a' to find the difference: \[c = a - b = (2n_1 + 1) - (2n_2 + 1)\]
04

Simplify the expression

Remove the parentheses and continue simplifying the expression: \[c = 2n_1 + 1 - 2n_2 - 1\] Now combine the constants and rearrange the terms: \[c = 2n_1 - 2n_2\]
05

Factor out the common factor

Factor out the common factor '2' from the expression: \[c = 2(n_1 - n_2)\]
06

Analyze the result

The difference 'c' is expressed as the product of two integers, 2 and \((n_1 - n_2)\). Since 2 is an even integer and the product of an even number with any integer (negative, positive or zero) is always even, the difference 'c' must always be even. So, we can conclude that the given statement is false: The difference of any two odd integers is not odd, but even.

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Key Concepts

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

Odd Integers
Odd integers are numbers that cannot be evenly divided by 2. When you divide an odd integer by 2, the remainder is always 1. Common examples of odd integers include 1, 3, 5, 7, and so forth. The mathematical expression that represents an odd integer is usually given by the formula \(2n + 1\), where \(n\) is an integer. This formula signifies that an odd number is always 1 more than an even number.
  • The form \(2n + 1\) can be broken down to show that any sequence of even numbers (multiples of 2) are added to 1 to yield an odd number.
  • Odd integers alternate with even integers on the number line.
  • Odd numbers maintain specific arithmetic properties, such as their behavior in addition and subtraction.
Recognizing the general form and properties of odd numbers is crucial when proving statements about sequences or patterns involving integers.
Integer Properties
Integer properties revolve around the familiar numbers used for counting and operations in everyday mathematics. These include whole numbers and their opposites, both positive and negative. Integers are key in understanding numeric relationships, particularly in operations like addition, subtraction, multiplication, and division. Each integer can be classified as either even or odd.
  • An even integer can be expressed as \(2n\), indicating divisible by the base unit of 2 without any remainder.
  • Odd integers, as discussed, are expressed as \(2n + 1\).
  • The following integer properties are essential: closure, commutative, associative, distributive, and identity properties, each relevant to operations within the integers.
Recognizing these properties enables deeper insight into more complex mathematical principles, especially when proving statements related to characteristics of even and odd integers.
Mathematical Proofs
Mathematical proofs are logical arguments designed to demonstrate the truth of a statement based on previously established facts and axioms. In mathematics, several types of proofs exist, including direct proofs, indirect proofs, and proof by contradiction. The proof provided in the original exercise is a form of direct proof.
  • In the exercise, we directly derive the expression for the difference between two odd integers using their general forms.
  • By simplifying the expression and analyzing the result, we identified the truth that the difference is even, revealing the error in the given statement.
  • This method ensures that each step logically follows from the previous one, verifying each transition with established mathematical properties.
Understanding how to construct and analyze mathematical proofs is essential in formally validating mathematical statements, ensuring that claims are justified with rigor and precision.

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Most popular questions from this chapter

The difference of any even integer minus any odd integer is odd.

Prove that the product of any two consecutive integers is even.

"Proof: Suppose \(r\) and \(s\) are rational numbers. By definition of rational, \(r=a / b\) for some integers \(a\) and \(b\) with \(b \neq 0\), and \(s=a / b\) for some integers \(a\) and \(b\) with \(b \neq 0\). Then \(r+s=a / b+a / b=2 a / b\). Let \(p=2 a\). Then \(p\) is an integer since it is a product of integers. Hence \(r+s=p / b\), where \(p\) and \(b\) are integers and \(b \neq 0\). Thus \(r+s\) is a rational number by definition of rational. This is what was to be shown."

Prove that for all positive integers \(a\) and \(b, a \mid b\) if, and only if, \(\operatorname{gcd}(a, b)=a\). (Note that to prove " \(A\) if, and only if, \(B, "\) you need to prove "if \(A\) then \(B\) " and "if \(B\) then \(A . "\) ")

If \(m\) and \(n\) are perfect squares, then \(m+n+2 \sqrt{m n}\) is also a perfect square. Why?
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