It is very similar to the proof that shows for all d>0 where d is a common divisor of that if and only if for some
Prove that c is negative if .
Suppose for some
Any common divisor d of a,b divides
Since d|c we know that |d| is less than or equal to |c|. This means that gcd(a,b)=|c|<gcd(a,b). This means that c must be -gcd(a,b), and since gcd(a,b) is always positive, then c must be negative.
Where gcd = greatest common divisor
|c|= magnitude of c
I am not guaranteeing its correctness but this is the basic idea... you may have to be more descriptive in areas.