Say if there is an element in $\Bbb Z_{900}$ with 30 as additive and multiplicative order

Question

I have a ring $\Bbb Z_{900}$ and I need to check if there is an element which has multiplicative and additive order 30. I think that is sufficient to check if the following system has at least one solution:

$$\begin{cases} \begin{aligned} 30x&\equiv0\ (\text{mod}\ 900) \\ x^{30}&\equiv1\ (\text{mod}\ 900) \end{aligned} \end{cases}$$

If my attempt is correct, how can I solve the system? The first congruence is easy to solve. How can I solve the second? Is there a method to reduce the power?

Answer 1

$30x \equiv 0 \, \, (\text{mod } 900) \Rightarrow 30x = 900z \ni z \in \mathbb{N} \Rightarrow x = 30z$. Now then $(30z)^3 = 30^3z^3 = 0 \, \, (\text{mod } 900)$. So there isn't any solution to that system.