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My investigation is about the Phi Function ?. I am investigating the different ways on how to find the Phi Functions of different numbers and finding easier ways of finding the Phi Functions of large numbers. I will go through four parts for this coursework. I will start from the simplest cases of numbers and will go to more complicated. 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: So ?(12)=4, because the positive integers less than 10 which have no factors other than 1, in common with 12 are 1, 5, 7, 11 i.e. 4 of them. These four numbers are not factors of 12. Also ?(6)=2, because the positive integers less than 6 which have no factors other than 1, in common with 6 are 1,...

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