Units of the ring $\mathbb{Z}_{12}$ I need to find the list of all units in the ring $\mathbb Z_{12}$.
I know that there must be a multiplicative inverse, $a_{-1}$, for there to be units, $a$, and that 
$$a\cdot a^{-1} = a^{-1} \cdot a = 1.$$
I think the multiplicative inverse here is $1$ and the answer tells me that the units are $\{1, 5, 7, 11\}$, but I am having trouble understanding.
 A: The multiplicative identity is $1$, as (I think) you meant. Each number is allowed to have its own inverse, so we check. $1$ clearly divides itself, so $1$ is always a unit. $5 \cdot 5 = 25 = 1$, so we see that $5$ is a unit. $7 \cdot 7 = 49 = 1$, so $7$ is a unit. And $11 \cdot 11 = 121 = 1$, so it's also a unit.
You might have noticed a few things - we happen to have squared each of these numbers, so that they are each their own inverse. Also, the primes that divide $12$ are $2$ and $3$, and $5,7,11$ are exactly the numbers that are coprime to $12$. I don't know what you've learned so far, but neither of these are coincidences (really, it's a multiplicative group now too, so perhaps your group theory can kick in).
A: Suppose that $xy = 1\pmod{12}.\:$ Then $\:3\ |\ 12\ |\ xy-1$ and this implies $x$ is coprime to $3\:$ (else $3\ |\ x, xy-1\ \Rightarrow\ 3\ |xy-(xy-1) = 1,\:$ contradiction). Similarly $x$ is coprime to $2$. This leaves only $\{\pm1,\:\pm5\}\pmod{12}$ as potential units. $\:\pm1$ are units. Ditto for $\pm5$ since $5^2\equiv 1\pmod{12}.$
Generally the Bezout identity for the GCD implies $n$ is a unit mod $m$ iff $\gcd(n,m)=1$. In a finite ring an element $a$ is a unit iff it is not a zero-divisor since the map $\:x\mapsto ax$ is onto iff it is $1$-to-$1$.
A: Another way of looking at it.  $(a,b)=1$ iff we have that there are integers $x,y$ with:
$$ ax+by =1$$ 
This implies that x is the multiplicative inverse of a , modb.
In our case , b =12 and $\{1,5,7,11 \} $ are all possible values for $a$:
$$ 1x +12\cdot 0 =1 ; 5 \cdot 5 + 12 \cdot (-2) =1 ; 7 \cdot 7 + 12 \cdot (-4)
 11 \cdot 11 + 12 \cdot (-10)=1$$
This means each of $\{ 1,5,7,11  \}$ are their own inverses in the ring $ \mathbb Z_{12} $
