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## Homework Statement

1) What are the possible values of [tex]m^{2}[/tex] + [tex]n^{2}[/tex] modulo 4?

2) Let [tex]d_{1}(n)[/tex] denote the last digit of n (the units digit)

a) What are the possible values of [tex]d_{1}(n^{2})[/tex]?

b) If [tex]d_{1}(n^{2})=d_{1}(m^{2})[/tex], how are [tex]d_{1}(n)[/tex] and [tex]d_{1}(m)[/tex] related?

3) a) Find all possible additive orders for a (mod 78)

b) for each order d in part a), find a number n with order d.

## The Attempt at a Solution

1) Would this just be looking for the equivalence classes. So in this case the equivalence classes would be [0], [1], [2], [3]?

2) I honestly have no idea what this question is even asking.

3) a) Well if a was a number given I could rewrite the problem as ax [tex]\cong[/tex]0 (mod 78). Where do I go from here in this case though?

b) Maybe this part will make more sense after I figure out a)