Problem 22 \(\forall n \in \mathbf{Z}\), if... [FREE SOLUTION]

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

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.

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.

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.

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.

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.

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Short Answer

Step by step solution

Recall the definition of prime numbers

Recall the properties of even and odd numbers

Analyze the case when n = 2

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

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

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Applied Mathematics

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.