Proving addition and multiplication

Question

(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:

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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)

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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.

Answer 1

Here are the hints for those problems:

1. 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.
2. 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 $a \cdot 1 = 1 \cdot a = a \bmod n$.
3. Follow the similar steps for this problem as for the first problem.
4. 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.