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One of the defining properties of 0 is that anything multiplied by it results in 0.
So in your operation, without being given the actual result, I'd say no, the question is ill-defined.
Then maybe there could be a different kind of zero that works normally with normal numbers, but when multiplied by some bizarre number gives a non-zero answer. Maybe after a few centuries we’ll be using bizarre numbers for computing weird stuff we can’t even imagine yet.
At the moment, that’s a solution looking for a problem, but give is time.
That sounds like you are describing a function.
You can have a function where inputting 0 gives a 0 output except certain conditions where an input of 0 gives non zero output. For example y=sin(x) gives 0 every time x=2π. Otherwise it ouputs a non zero number.