Finite set of congruences Is it true that for every $c$ there is a finite set of congruences 
$a_i(mod\,\,n_i) , c = n_1<n_2<n_3<...........<n_k  \,\,\,    (1)\\
$
So that every integer satisfies at least one of the congruence (1)

You are referring to Covering Systems of congruences. The link, and the name, will let you explore what is a quite large literature. You may also want to look at this survey by Carl Pomerance.  If the $n_i$ are strictly increasing, they cannot be chosen arbitrarily. 
