show numbers (mod $p$) are distict and nonzero Let's start with the nonzero numbers, mod $p$,
$1$, $2$, $\cdots$, $(p-1)$,
and multiply them all by a nonzero $a$ (mod $p$).
Notice that if we multiply again by the inverse of $a$ (mod $p$) we get back the numbers 
$1$, $2$, $\cdots$, $(p-1)$.
But my question is how the above process show that the numbers
$a\cdot 1$ mod $p$, $a\cdot 2$ mod $p$, $\cdots$, $a\cdot (p-1)$ mod $p$
are distinct and nonzero?
Thanks in advance. 

The set you are considering is really a field: $\mathbb{Z}/p\mathbb{Z}$. Since it is a field, it is closed under multiplication, therefore when you multipliy them by a nonzero element $a$, all of them are distinct and nonzero (but they are somehow permuted)
