Proof that integers (except $1$ and $-1$) don't have an integer multiplicative inverse This is a very elementary question, so please forgive me, but precisely because of this reason I haven't found an answer elsewhere:

How can it be proven that, among the integers, only $1$ and $-1$ have an integer multiplicative inverse? Or is that statement taken as an axiom?
Well, let's assume $ab=1$ when $a,b\in\mathbb{Z}$. Of course it means that both $a$ and $b$ are non zero. Because they are integers we can conclude that their absolute values are at least $1$. And then $1=|ab|=|a||b|\geq |a|$, and in the same way $1\geq |b|$. So both $a$ and $b$ must belong to the set $\{-1,1\}$.
As we are interested in 1/a which is just a symbol being adopted to denote the inverse of an element in Z.  Further identity element under multiplication in Z is 1. So if any member say ''a' of Z is invertible then there must be a member in Z denoted as 1/a or a^(-1) such that
 a.  a^(-1) =1.
 Now it's a simple algebra that in Z 1/a is an integer iff a= 1 or -1.
Hope it will help you.