1. ## Rings

Hi, The following is a challenge question that was set, which i do not know how to do;

The following is the defintion that was given to us:

Can anyone do this problem.

2. Hi

Let $m=2^ap_1^{a_1}...p_r^{a_r}$ a positive integer, $p_1,...,p_r$ distincts even primes. The group of units of the ring $(\mathbb{Z}_m,+,\times)$ is isomorphic to $(G\times\prod\limits_{i=1}^{r}\mathbb{Z}_{\phi(p_i ^{a_i})},+)$ where $\phi(p_i^{a_i})=p_i^{a_i-1}(p_i-1)$ and $G$ is:

$\{0\}$ if $a=0,1$
$\mathbb{Z}_2$ if $a=2$
$\mathbb{Z}_2\times\mathbb{Z}_{2^{a-2}}$ if $a\geq 3$

If I'm not wrong, that should be sufficient to prove that $m=2,3,4,6,8,12,24$ are the only solutions; but I let you show this, or disprove it

3. Thanks for the help;

I should be able to slove this problem know.