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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.

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