/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 \(\forall n \in \mathbf{Z}\), if... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 22

\(\forall n \in \mathbf{Z}\), if \(n\) is prime then \(n\) is odd or \(n=2\).

Short Answer

Expert verified
We have shown that for every integer n, if n is a prime number, then n must be odd or n equals 2. To prove this, we first analyzed the case when n = 2 and then showed that any even number greater than 2 cannot be a prime number. Therefore, if n is a prime number other than 2, it must be odd.

Step by step solution

01

Recall the definition of prime numbers

A prime number is an integer greater than 1 that has only two distinct positive divisors: 1 and itself. In other words, a prime number cannot be formed by multiplying two smaller positive integers.
02

Recall the properties of even and odd numbers

An even number is an integer that can be written in the form \(2n\) where n can be any integer, meaning it can be divided by 2 without any remainder. On the other hand, an odd number is an integer that cannot be divided by 2 without any remainder.
03

Analyze the case when n = 2

As given, we must show that if n is prime then n must be odd or n equals 2. If n = 2, then it has only two distinct positive divisors: 1 and 2. Therefore, 2 is a prime number and we have satisfied the given condition.
04

Exclude the case when n is an even number greater than 2

Let's assume that n is an even number greater than 2. Then it can be written in the form \(n = 2k\) where k is an integer greater than 1. In this case, n has at least three distinct positive divisors: 1, 2, and k. Since n has more than two divisors, n cannot be a prime number by definition.
05

Prove that if n is prime, n must be odd (except when n = 2)

We have already proven that when n = 2, it satisfies the given condition. Also, we have shown that if n is an even number greater than 2, it cannot be a prime number. Therefore, it's clear that if n is prime, it must not be an even number (except when n = 2). Consequently, n must be an odd number.
06

Conclusion

By analyzing the properties of prime numbers and odd and even numbers, we have shown that for every integer n, if n is a prime number, then n must be odd or n equals 2. This completes the proof.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Indicate whether the arguments in \(21-26\) are valid or invalid. Support your answers by drawing diagrams. No vegetarians eat meat. All vegans are vegetarian. No vegans eat meat.

Rewrite each of the following statements in the form \(" v\) \(x\), if then \(x\) and \(y\), if then \("\) or " a. All Java programs have at least 5 lines. b. Any valid argument with true premises has a true conclusion. c. The sum of any two even integers is even. d. The product of any two odd integers is odd.

Let \(D\) be the set of all students at your school, and let \(M(s)\) be "s is a math major," let \(C(s)\) be " \(s\) is a computer science student," and let \(E(s)\) be " \(s\) is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates \(M(s), C(s)\), and \(E(s)\). a. There is an engineering student who is a math major. b. Every computer science student is an engineering student. c. No computer science students are engineering students. d. Some computer science students are also math majors. e. Some computer science students are engineering students and some are not.

The following statement is true: " \(\forall\) nonzero numbers \(x, \exists\) a real number \(y\) such that \(x y=1 . .\) For each \(x\) given below, find a \(y\) to make the predicate " \(x y=1\) " true, a. \(x=2\) b. \(x=-1\) c. \(x=3 / 4\)

Consider the statement "There are no simple solutions to life's problems." Write an informal negation for the statement, and then write the statement formally using quantifiers and variables.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.