Proving addition and multiplication (1)Show that addition and multiplication mod n are associative operations.
(2)Show that there are both an additive and a multiplicative identity.
(3)Show that multiplication distributes over addition modulo n.
(4)Show that for every integer a there is an additive inverse -a:

I know, for both above, they mean that, 
(1) (a+b)+c ≡ a+(b+c) (mod n)
(ab)c ≡ ba (mod n)
(2) a+0 ≡ a (mod n)
    a*1 ≡ a (mod n)
(3) a(b+c) ≡ ab+ac (mod n)
(4) a+(-a) ≡ ab+ac (mod n)

Professor said not to worry about the proofs for these, but I'd rather know it for future courses.��
This seems easy to grasp, for the lack of a better word, but I do not know how I would prove this since class just started. 
 A: Here are the hints for those problems:


*

*Consider $a \bmod n$, $b \bmod n$ and $c \bmod n$.  Write those congruences in the form $z + nk$ where $k$ is any positive integers.  Add the first two congruences altogether and then, add that by $c \bmod n$.  Do the same for the right-hand side.  What do you know about $(a + b) + c \bmod n$ and $a + (b + c) \bmod n$.  Proving multiplicative associativity follows similarly as proving additive associativity.

*Additive identity is $0$, whereas multiplicative identity is $1$.  To show there exists an identity, prove that in $\mathbb{Z}_n$, the set of integers modulo $n$, for any $a \in \mathbb{Z}_n$, $a + 0 = 0 + a = a \bmod n$ and $a \cdot 1 = 1 \cdot a = a \bmod n$.

*Follow the similar steps for this problem as for the first problem.

*If there exists an additive inverse, then for any $a \in \mathbb{Z}_n$, there exists $c \in \mathbb{Z}_n$ such that $a + c = c + a = 0$.  Here $c = -a$.  I want you to prove this, starting with first similar steps for this problem also.

A: The main task is to show that these operations are well defined in the first place. The properties $(1)$–$(4)$ then follow by inspection.
Let an $n\geq1$ be given once and for all. Call two integers $x$ and $y$ equivalent if $x-y$ is an integer  multiple of $n$:
$$x\sim y\quad:\Longleftrightarrow\quad \exists j\in{\Bbb Z}:\quad x-y=j\>n\ .$$
Check that this is an equivalence relation. Denote the set of equivalence classes by ${\Bbb Z}_n$, and denote the equivalence class containing the number $x\in{\Bbb Z}$ by $[x]$. Define addition and multiplication of classes via representants, as follows:
$$[x]\oplus[y]:=[x+y],\qquad[x]\odot[y]:=[x\cdot y]\ .\tag{1}$$
One now has to check that the operations $\oplus$ and $\odot$ are well defined by $(1)$. This means that one has to verify that $[x]=[x']$ and $[y]=[y']$ together imply
$$[x+y]=[x'+y'],\quad [x\cdot y]=[x'\cdot y']\ .$$
Once this is accomplished the laws $(1)$–$(4)$ for ${\Bbb Z}_n$ immediately follow from the corresponding laws in ${\Bbb Z}$. It suffices to verify this for the first part of $(1)$:
$$\bigl([x]+[y]\bigr)+[z]=[x+y]+[z]=[x+y+z]=[x]+[y+z]=[x]+\bigl([y]+[z]\bigr)\ .$$
