   Chapter 9.3, Problem 41ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# For 40 and 41, use the definition of the Euler phi function φ from Section 7.1, exercises 51-53. Use the inclusion/exculsion principle to prove the folloeing: If n = p q r , where p , q , and r are distinct prime numbers, then φ ( n ) = ( p − 1 ) ( q − 1 ) ( r − 1 ) .

To determine

To prove the following: if n=pqr, where p,q and r are distinct prime numbers, then φ(n)=(p1)(q1)(r1) by using the inclusion/exclusion priniciple.

Explanation

Given information:

The inclusion/Exclusion Rule for two or three sets.

Concept used:

N(ABC)=N(A)+N(B)+N(C)N(AB)N(AC)N(BC)+N(ABC)

Calculation:

If n=p.q.r where p,q and r are distinct prime number.

By definition of Euler phi functions Φ, if p is a prime number and n is an integer with n1, then

Φ(pn)=pnpn1=pn(11p)

Let A,B and C be the set of all integers that are divisible by p,q and r respectively

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