This question comes from A Classical Introduction to Modern NT (Ireland & Rosen), chapter $9$ $\#17$. Note that $D$ is the set of Eisenstein integers.
Exercise 9.17 An element $x \in D$ is called primary if $x \equiv 2 \pmod 3$. If $x$ and $y$ are primary. show that $-xy$ is primary. If $x$ is primary, show that $x=\pm x_1 x_2 \cdots x_t$, where the $x_i$ are (not necessarily distinct) primary primes.
The first part is simple to prove: If $x \equiv y \equiv 2 \pmod 3$, then $-xy \equiv -2 \cdot 2 \equiv 2 \pmod 3$, which is primary.
How does the 2nd part of my proof look?
If $t\geqslant 3$ is odd, then we write $x_i \equiv 2 \pmod 3$ \begin{align*} x &= x_1 x_2 \cdots x_t\\ &= (2\cdot 2) \cdots (2\cdot 2) \cdot 2\\ &\equiv 1 \cdots 1 \cdot 2 \pmod 3\\ &\equiv 2 \pmod 3. \end{align*} If $t\geqslant 2$ is even, then we write $x_i \equiv 2 \pmod 3$ \begin{align*} x &= x_1 x_2 \cdots x_t\\ &= (2\cdot 2) \cdots (2\cdot 2)\\ &\equiv 1 \cdots 1 \pmod 3\\ &\equiv 1 \equiv -2 \pmod 3. \end{align*} Therefore, $x =\pm x_1 x_2 \cdots x_t$.
Is my proof correct? Am I allowed to assume the end and work backwards? I think that is more or less what I have done.