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eger n such that (n) = 14. %3D Euler phi-function for producing the term. Euler phi-function for producing the term

eger n such that (n) = 14. %3D Euler phi-function for producing the term. Euler phi-function for producing the term

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Using the Euler- Phi Function how can I solve 7.1 Question 8

the equation o (n) = 8 implies that no prime exceeding 9 divides n (otherWise oun P:-1> 8). Furthermore,7 cannot divide n because ifit did. 7-1=6 would be a factor of (n). It follows that n = 2ª3b5¢, where a. b. and c are nonnegative integers. We can also conclude that b = 0 or b= 1 and that e=0 or c - : otherwise, 3 or 5 would divide %3D To find all solutions, we need only consider four cases, When b =c = 0, we have n= 2", where a > 1. This implies that o(n) = 2a-1 which means that a =4 and n= When b=0 and c= 1, we have n = 29. 5. where a> 1 This implies that o (n) = 2".4, = 16. %3D %3D so a = 2 and n=20. When b= 1 and c =0. we have = 2ª. 3. where a > 1. This implies that (n) = 2ª-1. 2=2ª, so a = 3 and n = 24. Finally, when b = 1 and c= 1, we have n= 24. 3. 5. We need to consider the case where a= 0, as well as the case where a 2 1. %3D %3D %3D When a = 0, we have n = 15, which is a solution because d(15) =8. When a I, we have o(n) = 2ª-1.2·4= 2a+2. This means that a = 1 and n = 30. Putting everything together, we see that all the solutions to o (n) = 8 are n = 15, 16, 20, 24, and 30. %3D %3D %3D 7.1 EXERCISES 1. Determine whether each of the following arithmetic functions is completely multiplicative. Prove your answers. a) ƒ (n) = 0 b) ƒ (n) = 2 c) f (n) =n/2 u 301 = (u) f (p h) f(n) = n" %3D %3D I u= (u)f ( e) f (n) = n² f) f (n)=n! %3D %3D 2. Find the value of the Euler phi-function at each of these integers. u = (u)f (! a) 100 c) 1001 e) 10! b) 256 d) 2 · 3 · 5 · 7 · 11 · 13 f) 20! 3. Show that o (5186) = 0(5187) = $ (5188). %3D 4. Find all positive integers n such that o (n) has each of these values. Be sure to prove that you have found all solutions. b) 2 c) 3 d) 4 5. Find all positive integers n such that (n) = 6. Be sure to prove that you have found all solutions. 6) Find all positive integers n such that (n) = 12. Be sure to prove that you have found all solutions. 7. Find all positive integers n such that (n)= 24. Be sure to prove that you have found all solutions. Show that there is no positive integer n such that (n) = 14. 9. Can you find a rule involving the Euler phi-function for producing the terms of the sequence 1, 2, 2, 4, 4, 4, 6, 8, 6, . . .? 10. Can you find a rule involving the Euler phi-function for producing the terms of the sequence 2, 3,0, 4, 0, 4, 0, 5, 0, . . .? I1. For which positive integers n does o(3n) = 3¢(n)?
Transcribed Image Text:the equation o (n) = 8 implies that no prime exceeding 9 divides n (otherWise oun P:-1> 8). Furthermore,7 cannot divide n because ifit did. 7-1=6 would be a factor of (n). It follows that n = 2ª3b5¢, where a. b. and c are nonnegative integers. We can also conclude that b = 0 or b= 1 and that e=0 or c - : otherwise, 3 or 5 would divide %3D To find all solutions, we need only consider four cases, When b =c = 0, we have n= 2", where a > 1. This implies that o(n) = 2a-1 which means that a =4 and n= When b=0 and c= 1, we have n = 29. 5. where a> 1 This implies that o (n) = 2".4, = 16. %3D %3D so a = 2 and n=20. When b= 1 and c =0. we have = 2ª. 3. where a > 1. This implies that (n) = 2ª-1. 2=2ª, so a = 3 and n = 24. Finally, when b = 1 and c= 1, we have n= 24. 3. 5. We need to consider the case where a= 0, as well as the case where a 2 1. %3D %3D %3D When a = 0, we have n = 15, which is a solution because d(15) =8. When a I, we have o(n) = 2ª-1.2·4= 2a+2. This means that a = 1 and n = 30. Putting everything together, we see that all the solutions to o (n) = 8 are n = 15, 16, 20, 24, and 30. %3D %3D %3D 7.1 EXERCISES 1. Determine whether each of the following arithmetic functions is completely multiplicative. Prove your answers. a) ƒ (n) = 0 b) ƒ (n) = 2 c) f (n) =n/2 u 301 = (u) f (p h) f(n) = n" %3D %3D I u= (u)f ( e) f (n) = n² f) f (n)=n! %3D %3D 2. Find the value of the Euler phi-function at each of these integers. u = (u)f (! a) 100 c) 1001 e) 10! b) 256 d) 2 · 3 · 5 · 7 · 11 · 13 f) 20! 3. Show that o (5186) = 0(5187) = $ (5188). %3D 4. Find all positive integers n such that o (n) has each of these values. Be sure to prove that you have found all solutions. b) 2 c) 3 d) 4 5. Find all positive integers n such that (n) = 6. Be sure to prove that you have found all solutions. 6) Find all positive integers n such that (n) = 12. Be sure to prove that you have found all solutions. 7. Find all positive integers n such that (n)= 24. Be sure to prove that you have found all solutions. Show that there is no positive integer n such that (n) = 14. 9. Can you find a rule involving the Euler phi-function for producing the terms of the sequence 1, 2, 2, 4, 4, 4, 6, 8, 6, . . .? 10. Can you find a rule involving the Euler phi-function for producing the terms of the sequence 2, 3,0, 4, 0, 4, 0, 5, 0, . . .? I1. For which positive integers n does o(3n) = 3¢(n)?
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