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The Phi Function For any positive integer n, the Phi Function ?(n) is defined as the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n. Part 1 (a) Find the value of: (I) ?(3) (ii) ?(8) (iii) ?(11) (iv) ?(24) (b) Obtain the Phi-Function for at least 5 positive integers of your own choice. (a) (I) ?(3): 1 1 2 1,2 3 1,3 3 = 1,2 The number 3 only has 2 positive integers they are the numbers 1 and 2. (ii) ?(8): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 8 = 1,3,5,7 There are 4 positive integers for the number 8 (iii) ?(11): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 10 1,2,5,10 11 1,11 11 = 1,2,3,4,5,6,7,8,9,10 The number 11 has 10 positive integers, they are shown above. (iv) ?(24): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 10 1,2,5,10 11 1,11 12 1,2,4,6,12 13 1,13 14 1,2,7,14 15 1,3,5,15 16 1,2,4,8,16 17...

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