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Chapter 3: Problem 45

The difference of any two odd integers is even.

Short Answer

Expert verified
The difference of two odd integers, represented as \((2n + 1) - (2m + 1)\), simplifies to \(2n - 2m\), which can be factored as \(2(n - m)\). As \(n - m\) is an integer, the expression \(2(n - m)\) represents an even integer, proving the statement that the difference of any two odd integers is even.

Step by step solution

01

Define two odd integers

Let's consider two odd integers. We can represent any odd integer as \(2n + 1\) or \(2m + 1\), where n and m are any integers.
02

Find the difference of the odd integers

Now, we'll find the difference between these two odd integers: \((2n + 1) - (2m + 1)\)
03

Simplify the expression

Simplify the expression by performing the subtraction operation: \(2n + 1 - 2m - 1\)
04

Rearrange the terms

Rearrange the terms to group constants and variables: \(2n - 2m\)
05

Factor out 2 from the expression

Factor 2 out of the expression: \(2(n - m)\)
06

Prove the statement

Since \(n - m\) is an integer (difference of two integers), the expression \(2(n - m)\) represents an even integer, because any number multiplied by 2 results in an even integer. Therefore, the difference of any two odd integers is 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. These are integers such as 1, 3, 5, 7, and so on. For a number to be odd, when you divide it by 2, it leaves a remainder of 1.
An odd number can be expressed in the form of \(2n + 1\), where \(n\) is an integer. This formula adds 1 to an even number \(2n\) making the result an odd number.
  • If \(n = 0\), the odd number is \(1\).
  • If \(n = 1\), the odd number is \(3\).
  • If \(n = 2\), the odd number is \(5\).
Understanding the structure of odd integers helps in many areas of mathematics, like identifying patterns or proving properties like the one presented in the problem. Knowing how to express them in a formula makes it easier to work with them in equations.
Integer subtraction
When we subtract one integer from another, we're simply finding how far apart the numbers are on the number line. Subtracting integers can result in either positive or negative numbers, depending on the order of subtraction.
Let's break down the subtraction of two integers using a general formula:
  • If given integers \(a\) and \(b\), the subtraction can be written as \(a - b\).
  • The result is positive if \(a\) is greater than \(b\).
  • The result is negative if \(a\) is less than \(b\).
  • If \(a\) and \(b\) are equal, the result is zero.
Subtraction is essential for proving properties about numbers, such as checking the difference of two odd integers. Using variables like \((2n + 1) - (2m + 1)\) helps illustrate the process algebraically, simplifying the understanding of such concepts.
Even integer
Even integers are numbers that can be divided by 2 with no remainder. Examples include 0, 2, 4, 6, and so on. These numbers conform to the general form \(2k\), where \(k\) is an integer. This formula shows that any integer multiplied by 2 will result in an even number.
  • If \(k = 0\), the even number is \(0\).
  • If \(k = 1\), the even number is \(2\).
  • If \(k = 2\), the even number is \(4\).
The concept of even integers is crucial because it forms the basis for many proofs and properties in mathematics, such as the statement that the difference of two odd integers is an even integer. Since multiplying any integer by 2 gives an even result, expressions like \(2(n - m)\) ensure the difference is even, highlighting the reliable and predictable nature of even numbers.

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Most popular questions from this chapter

Prove that \(\sqrt{2}+\sqrt{3}\) is irrational.

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 \(A . "\) ")

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