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Chapter 8: Problem 24

. For which \(n \in \mathbf{Z}^{+}\)is \(\phi(n)\) odd?

Short Answer

Expert verified
The values of \(n\) for which \(\phi(n)\) is odd are: \(n = 1\) and \(n = 2\).

Step by step solution

01

Identifying the properties of Euler's totient function

Euler's totient function \(\phi(n)\) counts the positive integers up to a given integer \(n\) that are coprime (have no common divisors other than 1) to \(n\). Recall that \(\phi(n)\) is even for all \(n>2\). This is due to the fact that if a number \(a\) is relatively prime to \(n\), then so is \(-a \mod n\). Hence the numbers are paired off and \(\phi(n)\) is even.
02

Finding when \(\phi(n)\) is odd

From Step 1, we know that when \(n > 2\), \(\phi(n)\) is even. Thus, the only remaining cases to check are when \(n = 1\) and \(n = 2\). It is easy to check that \(\phi(1) = 1\) and \(\phi(2) = 1\). These are indeed odd. So, for \(n > 2\), \(\phi(n)\) cannot be odd.

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Key Concepts

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

Coprime Integers
When we talk about numbers that are coprime, we're referring to a pair of numbers that have no other common divisors except for the number 1. This is a foundational concept in number theory, a branch of mathematics concerned with the properties and relationships of numbers, especially integers. For example, 8 and 15 are coprime since the only positive integer that evenly divides both of them is 1.

To determine if two numbers are coprime, one can perform the Euclidean algorithm to find their greatest common divisor (GCD). If the GCD is 1, the numbers are coprime. Coprime numbers play a significant role in various areas, including the workings of the Euler's totient function, which is used to count the number of coprime integers to a given number within a set of positive integers.

Number Theory
Number theory is often seen as pure mathematics, focusing on the study of positive integers and integer-valued functions. Questions about divisibility, the distribution of primes, and the solutions to equations like Diophantine equations form part of this fascinating area of mathematics. An essential tool in number theory is Euler's totient function, denoted as \(\phi(n)\).

This function is a count of the numbers up to \(n\) that are coprime to \(n\). It has various applications, including the RSA algorithm for public-key cryptography, which hinges on the properties of \(\phi(n)\) for prime factorization. Understanding the behavior of the totient function, such as why \(\phi(n)\) is usually even, is critical for delving into deeper number theory topics and cryptography.

Positive Integers
Positive integers, also called natural numbers or counting numbers, are the numbers we use in everyday counting: 1, 2, 3, and so on. They are the building blocks of number theory and are inherently related to many mathematical concepts, including factors and multiples, prime numbers, and, of course, Euler's totient function. The totient function \(\phi(n)\) specifically looks at positive integers that are coprime to a given positive integer \(n\).

Understanding the nature of positive integers helps us systematically study mathematical structures and solve equations. It's interesting that for positive integers greater than 2, the value of \(\phi(n)\) somehow always turns out to be even, a peculiarity that can be related back to the fundamental properties of numbers and coprimality.

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Most popular questions from this chapter

Four applicants for a job are to be interviewed for 30 minutes each: 15 minutes with each of supervisors Nancy and Yolanda. (The interviews are in separate rooms, and interviewing starts at 9:00 A.M.) (a) In how many ways can these interviews be scheduled during a one-hour period? (b) One applicant, named Josephine, arrives at 9:00 A.M. What is the probability that she will have her two interviews one after the other? (c) Regina, another applicant, arrives at 9:00 A.M. and hopes to be finished in time to leave by \(9: 50 \mathrm{~A} . \mathrm{M}\). for another appointment. What is the probability that Regina will be able to leave on time?

If an arrangement of the letters in SURREPTITIOUS is selected at random, what is the probability that it contains (a) (exactly) three pairs of consecutive identical letters? (b) at most three pairs of consecutive identical letters?

2\. Establish the Principle of Inclusion and Exclusion by applying the Principle of Mathematical Induction to the number \(t\) of conditions.

Determine the number of integer solutions to \(x_{1}+x_{2}+\) \(x_{3}+x_{4}=19\) where \(-5 \leq x_{i} \leq 10\) for all \(1 \leq i \leq 4\)

For which positive integers \(n\) is \(\phi(n)\) a power of \(2 ?\)
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