Product of units modulo m Help!!! I can find units in a given $Z_m$ manually by testing out every single value which is relative to m,,, but in general I do not know a way how to find such units... hence I cant answer this question: Prove that the product of units mod m is congruent to $\pm 1$ mod m.
Any idea?

The product of all units mod $m$ can be written as $P=\prod_{x^2\ne1} x \prod_{y^2=1} y$.
The first product is $1$ because each $x$ is paired with $x^{-1}$ and they are different.
Therefore, $P=y_1 \cdots y_n$ with $y_i^2=1$.
If $n=1$, then $P=1$ because $y=1$ is certainly there.
If $n=2$, then $P=-1$ because $y=-1$ is certainly there.
If $n>2$, Then $y \mapsto -y$ is a permutation of the elements such $y^2=1$. This permutation has no fixed points, that is, $-y\ne y$. This implies than $n$ is even. Thus, every $y$ is paired with $-y\ne y$ and $y(-y)=-1$. Therefore, $P=(-1)^{n/2}=\pm 1$.
(Actually, in the last case, $n/2$ is even and $P=1$. This follows from Lagrange's theorem applied to a subgroup $\{1,-1,y,-y\}$.)
