This question popped up in a project I'm working on and it wasn't immediately familiar as a linear algebra problem I've seen before. So if this is a well known thing, apologies. Just point me in the right direction.
Suppose I have a set of linear equations $ax_{1}=C , bx_{2}=C, ...$ where {$a,b,c...$} are integers. It's of course easy to find $x_{1},x_{2}...$ etc.
What I would like to find is instead the set of all integers {$a', b', c'...$} such that $a'x_{1}+b'x_{2}+...=C$