# Maximum order of element in group of units in a ring

Let $s$ be a natural number and $U(s)$ be a group of units in the ring $\mathbb{Z}/s\mathbb{Z}$. Let $\phi(s) = 2^{k_1}p^{k_2}$, where $p$ is a an odd prime number. I don't understand why the maximum order of any element in $U(s)$ is $$lcm (2^{k_3},\phi(p^{k_2}))$$ and $k_3 = 0$, if $k_1\le0$ $k_3 = 1$, if $k_1=2$ and $k_3 = k_1-2$, if $k_1\ge3$

• What did you mean by $lcm(2^{k_3}p^{k_2})$? That number is the product, as $p$ is an odd prime. – Emre May 21 '16 at 13:47
• I am sorry, i just corrected it. – Mayank May 21 '16 at 13:49