1. ## Modulo Definition of Addition and Multiplication

Some help with the following proof would be great:

Zn is the set of integers modulo n

Fix an integer n >= 2. Addition and multiplication on Zn are commutative, associative, and distributive. Prove that the set Zn has an additive identity, a multiplicative identity, and additive inverses.

Addition and multiplication are defined in the following way. If a ≡ a' (mod n) and b ≡ b' (mod n) then

a+b ≡ a' + b' (mod n) and ab ≡ a'b' (mod n)

and so [a] + [b] = [a+b] and [a] * [b] = [ab]

Any help is appreciated!

2. So you want to show that there exist a number "b" such that, for any a, [a]+ [b]= [a].
Since [a]+ [b]= [a+ b], that means you want to show that there exist a number [a+ b]= [a] for any a. That is, of course, the same as saying that a+ b= a (mod n). What must b be equal to?

Similarly, you want to show that there exists a number "c" such that [ac]= [a]. And that is the same as saying that ac= a (modulo n). What must c be equal to?

To show that there exist additive inverses, you need to show that, for any a, there exist a number c such that [a+ c]= b (where b is the number you found in the first part). That is, you need to show that there exist a+ c= b (mod n).