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9 statements about equivalence:

Let $n$ be a positive integer and $a,b,c,d$ be integers. Provide a proof of the following statements.

1. $a\equiv a~(mod~n)$

2. if $a\equiv b~(mod~n)$, then $b\equiv a~(mod~n)$

3. if $a\equiv b~(mod~n)$ and $b\equiv c~(mod~n)$, then $a\equiv c~(mod~n)$

4. if $a\equiv b~(mod~n)$ then $a+c\equiv b+c~(mod~n)$

5. if $a\equiv b~(mod~n)$ then $a\cdot c\equiv b\cdot c~(mod~n)$

6. if $a\equiv b~(mod~n)$ and $c\equiv d~(mod~n)$, then $a+c\equiv b+d~(mod~n)$

7. if $a\equiv b~(mod~n)$ and $c\equiv d~(mod~n)$, then $a\cdot c\equiv b\cdot d~(mod~n)$

8. if $c \cdot a\equiv c \cdot b~(mod~n)$, then it is not necessarily the case that $a \equiv b~(mod~n)$

9. if $a\cdot b \equiv a\cdot c~(mod~n)$ and $gcd(a,n)= 1$, then $b\equiv c~(mod~n)$