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To me a free month means one entire calendar month or the equivalent will be provided at no charge. A half or partial month sounds to me like false advertising.
If I were the landlord and made such an offer (I wouldn't) I would make it clear that the free month is the last calendar month. I'm a "get the money as soon as possible" kind of businessman.
However landlord in this situation may mean the first month as inducement to move in with less money down. Offers such as this should come with a description of exactly what is being offered.