I assume that you do not want to re tile the basement. If you went this route, you could just go over this.
Since you want to paint the floor, you must get this stuff up. You will have quite a problem with this removal.
It will bleed through the paint and you definately cannot sand it. Find some sort of stripper that you can apply to this stuff and go from there.
My two cents worth is to re tile over this and be done with it.
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