It is a classic and fundamental result that the function is multiplicative. Meaning,
The proof in my book is not very elegant. Thus, I attempted to find my own proof. I need help with completing it however.
Proof (Incomplete): For any define a set . Thus is the set of all integers relatively prime to but not equal to . Now define a binary operation on this set as multiplication modulo . It can be show that forms a group. (Just a note this is the group we use to show the proof of Euler's Generalized Theorem of Fermat). Now this is the final step, let for some . Thus, is also a group. This is the incomplete step show that . The proof is complete because since both and are finite the cardinality of is . But has cardinality we finally have that
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