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Chapter 4: Problem 5

Show that if \(n\) is a positive integer, then \(\phi(2 n)=2 \phi(n)\) if \(n\) is even.

Short Answer

Expert verified
#Question# Show that if \(n\) is a positive integer, then \(\phi(2n)=2\phi(n)\) if \(n\) is even. Here, \(\phi\) denotes the Euler's totient function. #Answer# To show that, first rewrite \(n\) as \(n=2k\), where \(k\) is a positive integer. Then, using the property of Euler's totient function for relatively prime numbers, calculate \(\phi(2^2)\). Finally, substitute the calculated value and simplify the expression to obtain the result \(\phi(2n) = 2\phi(n)\).

Step by step solution

01

Rewrite \(n\) as \(n=2k\) where \(k\) is a positive integer

Since \(n\) is even, it can always be written as \(n=2k\), where \(k\) is a positive integer. Now, we'll use the property of the Euler's totient function for the given case:
02

Use \(\phi(mn)=\phi(m)\phi(n)\) property, if \(m\) and \(n\) are relatively prime

Since \(4=2^2\) and \(k\) are relatively prime, we can apply the given property: \(\phi(4k) = \phi(2^2) \cdot \phi(k)\) Next, we'll find the value of \(\phi(2^2)\):
03

Calculate \(\phi(2^2)\)

Euler's totient function for any prime power \(p^k\) is given by \(\phi(p^k) = p^k - p^{k-1}\). In this case, since \(2\) is a prime number, we have: \(\phi(2^2) = 2^2 - 2^{2-1} = 4 - 2 = 2\) Now, to find \(\phi(4k)\), we'll substitute the value of \(\phi(2^2)\) and simplify:
04

Substitute the value of \(\phi(2^2)\) and simplify

We have found \(\phi(2^2) = 2\), and we know \(\phi(4k) = \phi(2^2) \cdot \phi(k)\). Substituting the value and simplifying gives: \(\phi(4k) = 2 \cdot \phi(k)\) Lastly, we'll compare this result with the required result:
05

Compare the result

Our result is: \(\phi(4k) = 2 \cdot \phi(k)\) Since \(4k = 2 (2k)\) and \(n = 2k\), this result can be written as: \(\phi(2n) = 2 \cdot \phi(n)\) This is the required result, and we have successfully shown that if \(n\) is a positive integer, then \(\phi(2n) = 2\phi(n)\) if \(n\) is even.

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Key Concepts

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

Number Theory
Number theory is a fascinating branch of mathematics that focuses on the properties and relationships of numbers, particularly integers. It's an essential area of study that opens the door to understanding more complex concepts and ideas, including primes, divisibility, and modular arithmetic.

Some key concepts in number theory include:

  • Integers: These are whole numbers, including negative numbers, zero, and positive numbers.
  • Primes: Numbers greater than 1 that have no divisors other than 1 and themselves, such as 2, 3, 5, 7, etc.
  • Divisibility: A condition where one integer can be divided by another without leaving a remainder.

This area of math forms the foundation of various algorithms and cryptographic systems, making it not only theoretically interesting but also practically relevant.
Relatively Prime
When two numbers are referred to as "relatively prime," it means they share no common factors other than 1. This concept is crucial for applying certain mathematical functions and theorems. For instance, when finding the Euler's Totient Function, the property of being relatively prime is often used.

How to Determine if Two Numbers are Relatively Prime:
  • Find the GCD: Calculate the greatest common divisor (GCD) of the numbers. If the GCD is 1, the numbers are relatively prime.
  • Prime Factorization: Break down each number into its prime factors and see if they have any in common.

The use of relatively prime numbers is vital in computations involving modular arithmetic and cryptographic algorithms.
Prime Powers
Prime powers play a significant role in number theory, particularly in Euler's Totient Function. A prime power is an expression of the form \(p^k\), where \(p\) is a prime number and \(k\) is a positive integer. This concept helps in computing various mathematical functions easily.

Calculating Euler's Totient Function for Prime Powers:
  • The formula \(\phi(p^k) = p^k - p^{k-1}\) helps in determining the totient function of a prime power.
  • This formula reflects the idea that among the \(p^k\) numbers, \(p^{k-1}\) of them are not relatively prime with \(p^k\) as they are multiples of \(p\).

Understanding prime powers is essential when dealing with divisibility, factorization, and various theorems within number theory.
Positive Integer
A positive integer is an integer greater than zero. Positive integers are often the focus in many mathematical problems and concepts, including Euler's Totient Function and number theories like divisibility and prime analysis.

Key Characteristics of Positive Integers:
  • Natural Numbers: Positive integers start from 1 and go upwards (1, 2, 3, ...).
  • Counting Numbers: They are what we use in daily life to count objects or quantities.

In the context of Euler's Totient Function, positive integers are often the domain of study to assess the number of integers relatively prime to a given number, which is a critical aspect of various applications in mathematics.

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Most popular questions from this chapter

Define the arithmetic function \(g(n)\) by the following. \(\mathrm{g}(\mathrm{n})=1\) if \(n=1\) and 0 for \(n>1\). Prove that \(g(n)\) is multiplicative.

Use the Mobius inversion formula and the identity \(n=\sum_{d \mid n} \phi(n / d)\) to show that \(\phi\left(p^{t}\right)=p^{t}-p^{t-1}\) where \(p\) is a prime and \(t\) is a positive integer.

Find the sum of positive integer divisors and the number of positive integer divisors of 35

Show that if \(n\) is an odd integer, then \(\phi(4 n)=2 \phi(n)\).

Find the eighth perfect number.
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