Unsatisfiability of an Integer Programming Problem?

Question

Given a set of Diophantine equations of the form:

1. $(a_0A_0+a_1A_1+...a_nA_n)=c$
2. $c>0$

And all the remaining equations of the two form:

3. $a_i+a_j+..a_l=1$

where $a_1−a_n$, c are variables. And $A_1−A_n$ are constants.

Query: Does there exist a positive integer $c$, such that a non-negative integer solution to these equations does-not exist for that $c$?

I am aware the generic Integer Programming Problem (for satisfying argument) is NP-Complete. So, this in that sense, is the Dual to the problem. Thus, Is problem much simpler than the satisfiable case?

Answer 1

If the $A$s have a common factor that $c$ does not there will be no solution