Purchase Solution

# Proof of the Irrationality using Euler's phi-function

Not what you're looking for?

2. Apply the proof of the irrationality of sqrt(2) to a) sqrt(3) and b)
sqrt(4). If the proof breaks down, indicate precisely why.

3. Euler's phi-function is defined such that for n > 0, phi(n) = |{m <
n: gcd(m,n)=1}|. So, e.g., phi(4) = |{1,3}| = 2; phi(5) = |{1,2,3,4}| =

4.
a. Show that for prime p, phi(p) = p-1.

b. Show that for prime p and q, phi(p*q) = (p-1)*(q-1).

##### Solution Summary

Irrationalities and primes are manipulated. The proof of the irrationality using Euler's phi-functions.

##### Solution Preview

Please see the attached file for the full solution.

Thanks for using BrainMass.

2. Apply the proof of the irrationality of sqrt(2) to a) sqrt(3) and b)
sqrt(4). If the proof breaks down, indicate precisely why.

Solution:
You haven't indicated which proof to use so I hope you now this one:
a)Suppose
is rational. This means that for some integers p and q
Square both sides

Multiply both sides by q2

By Fundamental Theorem of Arithmetic, each of ...

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.