I think you're on the right track with this one. I have some reasoning now using that:
T exists if such a triple exists that
x=as, y=bs, z=cs
Where s (scale) must contain the LCM of a,b,c as a factor.
This is still quite new to me, so I can't be too sure about my reasoning. I don't know either if cases outside this can or can't exist.
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