# Solubility of a Diophantine equation [closed]

Is it true that for positive integers $$B$$ and $$G$$, the following equation in $$B_1,B_2,G_1,G_2$$ has a non-negative integer solution for any value $$c$$ such that $$|c|\le B G$$ :

$$G_1B_1-G_2B_2 = c$$

under the constraints $$G_1\le G$$, $$G_2\le G$$, $$B_1\le B$$, $$B_2\le B$$, $$G_1+G_2\ge G$$, $$B_1+B_2\ge B\,$$?

Although not the source of the problem, the geometric picture is that of existence of a triangle of area $$|c/2|$$ with non-negative integer coordinates pivoted by a vertex on the origin with sufficiently long arms, but is within the (axis-aligned) rectangle at origin with sides $$+B$$ and $$+G$$.

I did check computationally for $$B,G \le 100$$ and the assertion seems to be true.

If $$B$$ and $$G$$ are relatively prime then a solution exists with $$B_1+B_2 = B$$ and $$G_1 + G_2 = G$$. For in such case the equation reduces to $$G_1B+B_1G=c+BG$$, the LHS of which evidently can never be the same modulo $$BG$$ as we vary $$G_1$$ from $$1$$ to $$G$$ and $$B_1$$ likewise.

It is not clear or obvious to me why the statement should be true (or false) for cases where $$B$$ and $$G$$ are not coprime.