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All these are good pairs:

$$(0, 0), (A, B), (2A, 2B), (3A, 3B), \ldots \pmod{n}$$

But are there any other pairs?

actually it was a programming problem with $A,B,n \leq 10000$ but it seems to have a pure solution.

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    Can you please reformulate your question. Your title is not clear at all and as it stand true for any a, b.2012-02-28
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    @NickyHekster That was partially my fault, I changed some capitalizations because a couple were clearly wrong, but changed to many. However the question still makes no sense.2012-02-28
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    Is it unclear ?2012-02-28
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    @a-z unclear is one thing,if there is given (A,B) and you are looking for (a,b) then it seems that they are different and maybe (a,b) are some combination of (A,B) or others,so guys are asking to define exactly why do you need (a,b) when (A,B) are given2012-02-28
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    @AndréNicolas : But $(5,6)$ is not "other". take $k=7$ then $(kA,kB) = (5,6)$.2012-02-28
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    @a-z: OK, so you are automatically reducing modulo $n$, which is the reasonable thing to do.2012-02-28

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