* Looking at modular multiplication on Zn - Take the set of integers Z17 * - what are all the inverses of all elements in Z17 ?

Write the multiplication table to find the inverses by brute force, or use the division algorithm.

True, both would work, can you give an example of how you would do this? I'm looking at considering different ways of expressing the answer - Thanks

You can try little arithmetic tricks: the inverse of 1 is 1, the inverse of 2 is 9, the inverse of 3 is 6, the inverse of 4 = 2^2 is 4^(-1) = 2^(-2) = (2^(-1))^2 = 9^2 = 13 (everything here's done modulo 17, of course),
the inverse of 5 is 7, the inverse of 10 = 2*5 is 10^(-1) = 2^(-1)*5^(-1) = 9*7 = 12 , etc.
If your other option is write down multiplication tables I think the above can be shorter.

Tonio