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Chapter 1: Problem 3

Show that the square of an even number is an even number using a direct proof.

Short Answer

Expert verified
The square of any even number \( n = 2k \) is always even, as \( n^2 = 4k^2 \) which is divisible by 2.

Step by step solution

01

Define an even number

An even number can be expressed as any integer that is divisible by 2. Let's represent an even number as \( n = 2k \) where \( k \) is an integer.
02

Square the even number

To show that the square of an even number is also even, we first square the expression for an even number. The expression is \( n = 2k \). So, squaring both sides we get \( n^2 = (2k)^2 \).
03

Simplify the squared number

Simplify \( (2k)^2 \). It can be rewritten as \( (2k) \times (2k) = 4k^2 \).
04

Factor the result

Notice that \( 4k^2 \) can be factored to \( 2(2k^2) \). This shows that \( 4k^2 \) is clearly divisible by 2, indicating that \( 4k^2 \) is an even number.
05

Conclude the proof

Since \( n^2 = 4k^2 \) is an even number, we have shown that the square of an even number is indeed even.

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Key Concepts

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

Even Numbers
An even number is any integer that can be exactly divided by 2. To put it simply, even numbers are those numbers that end in 0, 2, 4, 6, or 8 when written in decimal form. Some examples include 2, 4, 6, 8, 10, and so on. If you express an even number algebraically, it can be written as \( n = 2k \), where \( k \) is an integer. This formula helps us to understand and manipulate even numbers in mathematical proofs and problems. By knowing this, we can easily identify and work with even numbers in various calculations and proofs.
Mathematical Proof
Mathematical proof is a logical series of statements leading from the assumptions to the conclusion. Direct proof is a type of mathematical proof. In direct proof, you start with what is given and use logical steps to arrive at the statement that needs to be proven.
Steps used in direct proof:
  • 1. State what is given and what needs to be proven.
  • 2. Introduce definitions relevant to the problem (e.g., even number).
  • 3. Use algebraic manipulations to show the statement is true.
  • 4. Conclude by stating that the original claim has been proven.
Using direct proofs can make showing mathematical concepts and relationships clear and straightforward.
Squaring Integers
Squaring an integer means multiplying the integer by itself. For example, if we square the integer 3, we get \( 3^2 = 9 \). When it comes to even numbers, we can use the expression for an even number, \( n = 2k \), to illustrate the concept. If we square an even number, like so:
\[ (2k)^2 = 4k^2 \]
We see that the result, \( 4k^2 \), is also a multiple of 2 (since \( 4 = 2 \times 2 \)). Therefore, \( 4k^2 \) is another even number. This proof shows that the square of an even number always results in another even number. Understanding this concept helps in building a solid foundation for more advanced mathematical operations and theorems.

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Most popular questions from this chapter

Prove that there are no solutions in integers \(x\) and \(y\) to the equation \(2 x^{2}+5 y^{2}=14 .\)

Suppose that the domain of the propositional function \(P(x)\) consists of the integers \(1,2,3,4,\) and \(5 .\) Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions. $$ \begin{array}{ll}{\text { a) } \quad \exists x P(x)} & {\text { b) } \forall x P(x)} \\ {\text { c) } \quad \neg \exists x P(x)} & {\text { d) } \neg \forall x P(x)}\end{array} $$ e) \(\quad \forall x((x \neq 3) \rightarrow P(x)) \vee \exists x \neg P(x)\)

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) No one is perfect. b) Not everyone is perfect. c) All your friends are perfect. d) At least one of your friends is perfect. e) Everyone is your friend and is perfect. f) Not everybody is your friend or someone is not perfect.

Use a proof by exhaustion to show that a tiling using dominoes of a \(4 \times 4\) checkerboard with opposite corners removed does not exist. [Hint: First show that you can assume that the squares in the upper left and lower right corners are removed. Number the squares of the original checkerboard from 1 to \(16,\) starting in the first row, moving right in this row, then starting in the leftmost square in the second row and moving right, and so on. Remove squares 1 and \(16 .\) To begin the proof, note that square 2 is covered either by a domino laid horizontally, which covers squares 2 and \(3,\) or vertically, which covers squares 2 and \(6 .\) Consider each of these cases separately, and work through all the subcases that arise. \(]\)

Find all squares, if they exist, on an \(8 \times 8\) checkerboard such that the board obtained by removing one of these squares can be tiled using straight triominoes. [Hint: First use arguments based on coloring and rotations to eliminate as many squares as possible from consideration. \(]\)
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