quick question

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when i buy a property from a seller in october.....are the january - september taxes due at closing...paid by the seller right? then i pick up the october - december taxes at the end of the year? how does this work? And if the seller has the taxes in with his mortgage put into escrow ...he/she uses this to pay off the taxes owed .....instead of the taxes owed comming out of what they would pocket at closing...does that make since?

thanks in advance