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I'm having some trouble with the following problem:

Considering an independent system of linear equations in $x$ and $y$ with integer coefficients \begin{equation} ax + by = c_1\\ cx + dy = c_2. \end{equation}

Suppose there does not exist an integer solution for $[c_1,c_2] = [p,0]$ and $[c_1,c_2] = [0,q]$, where $p,q\in\mathbb{Z}$. I would like to show that there does not exist an integer solution for $[c_1,c_2] = [p,q]$.

I already tried using matrix inverses, exploiting the independence of $[p,0]$ and $[0,q]$. However I could not find an answer. Any ideas on the matter are welcome, thanks in advance.

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  • $\begingroup$ Are $p,q$ particular fixed values, or do you assume the condition for all $p,q\in\mathbb{Z}$? If the former, consider the case $a=b=c=p=q=1, d=-1$. $\endgroup$ – Klaus Draeger Nov 19 '15 at 12:24
  • $\begingroup$ Ah, saw your solution just after posting my identical one. $\endgroup$ – Gerry Myerson Nov 19 '15 at 12:26
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$x+y=1,x-y=0$ has no integer solutions. $x+y=0,x-y=1$ has no integer solutions. $x+y=1,x-y=1$ has integer solutions.

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    $\begingroup$ Somehow I thought of this example, but must have made some (very) dumb mistake in adding the numbers. Thanks for your answer. :) $\endgroup$ – Tom Ultramelonman Nov 20 '15 at 22:39

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