Any finite group $G$ can be decomposed as a direct product of its cyclic groups. Let
$$G =\langle g_1 \rangle \times \langle g_2 \rangle \times \ldots \times \langle g_t \rangle$$
where the order of each $g_i$ is $c_i$ for $i \in \{1,...,i\}$. For any $g \in G$ one can write
$$g = (g_1^{\alpha_1},g_2^{\alpha_2},\ldots,g_t^{\alpha_t})$$
Where $\alpha_i \le c_i$ for all $i \in[t]$.
Questions : Is it true that $g = g_1^{\alpha_1}\cdot g_2^{\alpha_2} \cdot \ldots \cdot g_t^{\alpha_t} $ where $\cdot$ is the group operation of $G$?
I have tried for
$Z_{15} = Z_5 \times Z_3$.
Yes this is true. See for example the answer to A finite group is a product of cyclic groups?
For every $g_i$ you can write $g_i=\underbrace{e\cdots e}_{i-1\text{ times}}\cdot g_i\cdot\underbrace{e\cdots e}_{t-i\text{ times}}$ and this immediately implies your decomposition.