Suppose you have an equation like $ax_1+ax_2+\dots +ax_k=c$ with integers $a_i$. What does the solution set is given by a form of $p+\mathbb Zq_1+\dots+\mathbb Zq_{k-1}$ where $q_i\in\mathbb Z^k$ only depends on $a_1,a_2,\dots$ and $ p\in \mathbb Z^k$ mean?
So what is the defintion of $\mathbb Z q_i$? Is this $c\cdot (q_{1,i},q_{2,i},\dots , q_{k,i})^T$ for some $c\in \mathbb Z$
The notation $\Bbb Zq$ denotes the integral multiples of $q$, i.e., the set $$ \Bbb Zq=\{0,\pm q,\pm 2q,\pm 3q ,\cdots \}. $$