Because of a problem I ran into I am trying to get a quick start in covering with arithmetic progressions.

First I want to say I am aware of this previously asked question: Covering $\mathbb{N}$ with prime arithmetic progressions

Similarly to what is asked there I am interesting in covering with arithmetic progressions of the type $A_{i}=k_{i}+np_{i}$.

Differently, my interests are however in covering of finite sets of the type $\{1, 2, \ldots, N\} \subset \mathbb{N}$ where $k_{i}<p_{i}$ and $p_{1} \leq p_{i} \leq p_{m}$.

By checking out some numbers it looks like that if all $k_{i}=p_{i}-1$ the covered set is $\{1, 2, \ldots, p_{m+1}-2 \}$.

I would appreciate all relevant information possible, like: Has this been proved?, Then by whom?; Will any set of values $k_{i}<p_{i}-1$ give less coverage?; Any textbooks discussing this or very similar kind of problems; Websites; etc.