Using the Euler- Phi Function how can I solve 7.1 Question 2 the equation o (n) = 8 implies that no prime exceeding 9 divides n (otherWise ounP:-1 8). Furthermore,7 cannot divide n because ifit did. 7-1=6 would be a factorof (n). It follows that n = 2ª3b5¢, where a. b. and c are nonnegative integers. We canalso conclude that b = 0 or b= 1 and that e=0 or c - : otherwise, 3 or 5 would divide%3DTo find all solutions, we need only consider four cases, When b =c = 0, we haven= 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%3Dso a = 2 and n=20. When b= 1 and c =0. we have = 2ª. 3. where a 1. This impliesthat (n) = 2ª-1. 2=2ª, so a = 3 and n = 24. Finally, when b = 1 and c= 1, we haven= 24. 3. 5. We need to consider the case where a= 0, as well as the case where a 2 1.%3D%3D%3DWhen a = 0, we have n = 15, which is a solution because d(15) =8. When a I, wehave o(n) = 2ª-1.2·4= 2a+2. This means that a = 1 and n = 30. Putting everythingtogether, we see that all the solutions to o (n) = 8 are n = 15, 16, 20, 24, and 30.%3D%3D%3D7.1 EXERCISES1. Determine whether each of the following arithmetic functions is completely multiplicative.Prove your answers.a) ƒ (n) = 0b) ƒ (n) = 2c) f (n) =n/2u 301 = (u) f (ph) f(n) = n"%3D%3DI u= (u)f (e) f (n) = n²f) f (n)=n!%3D%3D2. Find the value of the Euler phi-function at each of these integers.u = (u)f (!a) 100c) 1001e) 10!b) 256d) 2 · 3 · 5 · 7 · 11 · 13f) 20!3. Show that o (5186) = 0(5187) = $ (5188).%3D4. Find all positive integers n such that o (n) has each of these values. Be sure to prove that youhave found all solutions.b) 2c) 3d) 45. Find all positive integers n such that (n) = 6. Be sure to prove that you have found allsolutions.6) Find all positive integers n such that (n) = 12. Be sure to prove that you have found allsolutions.7. Find all positive integers n such that (n)= 24. Be sure to prove that you have found allsolutions.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 sequence1, 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 sequence2, 3,0, 4, 0, 4, 0, 5, 0, . . .?I1. For which positive integers n does o(3n) = 3¢(n)?

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Using the Euler- Phi Function how can I solve 7.1 Question 2 the equation o (n) = 8 implies that no prime exceeding 9 divides n (otherWise ounP:-1> 8). Furthermore,7 cannot divide n because ifit did. 7-1=6 would be a factorof (n). It follows that n = 2ª3b5¢, where a. b. and c are nonnegative integers. We canalso conclude that b = 0 or b= 1 and that e=0 or c - : otherwise, 3 or 5 would divide%3DTo find all solutions, we need only consider four cases, When b =c = 0, we haven= 2&#34;, 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&#34;.4,= 16.%3D%3Dso a = 2 and n=20. When b= 1 and c =0. we have = 2ª. 3. where a > 1. This impliesthat (n) = 2ª-1. 2=2ª, so a = 3 and n = 24. Finally, when b = 1 and c= 1, we haven= 24. 3. 5. We need to consider the case where a= 0, as well as the case where a 2 1.%3D%3D%3DWhen a = 0, we have n = 15, which is a solution because d(15) =8. When a I, wehave o(n) = 2ª-1.2·4= 2a+2. This means that a = 1 and n = 30. Putting everythingtogether, we see that all the solutions to o (n) = 8 are n = 15, 16, 20, 24, and 30.%3D%3D%3D7.1 EXERCISES1. Determine whether each of the following arithmetic functions is completely multiplicative.Prove your answers.a) ƒ (n) = 0b) ƒ (n) = 2c) f (n) =n/2u 301 = (u) f (ph) f(n) = n&#34;%3D%3DI u= (u)f (e) f (n) = n²f) f (n)=n!%3D%3D2. Find the value of the Euler phi-function at each of these integers.u = (u)f (!a) 100c) 1001e) 10!b) 256d) 2 · 3 · 5 · 7 · 11 · 13f) 20!3. Show that o (5186) = 0(5187) = $ (5188).%3D4. Find all positive integers n such that o (n) has each of these values. Be sure to prove that youhave found all solutions.b) 2c) 3d) 45. Find all positive integers n such that (n) = 6. Be sure to prove that you have found allsolutions.6) Find all positive integers n such that (n) = 12. Be sure to prove that you have found allsolutions.7. Find all positive integers n such that (n)= 24. Be sure to prove that you have found allsolutions.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 sequence1, 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 sequence2, 3,0, 4, 0, 4, 0, 5, 0, . . .?I1. For which positive integers n does o(3n) = 3¢(n)?

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2. Find the value of the Euler phi-function at each of these integers- e) 10! b) 256 d) 2.3-5-7-11- 13 f) 20!