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Your landlord must give you a written Eviction Notice, sometimes called a Notice To Quit.. If you do not have a lease, the Notice will tell you that you have either 7 days or 30 days to move out. (An oral eviction notice is generally not legal). Keep your eviction notice. The Eviction Notice is not the same thing as a court order.
Unless you agree to move out, the new owner cannot evict you without bringing an eviction proceeding in Housing Court. There is no pre se rule as to how long you have to stay. However, as a month-to-month tenant, 30 days is ordinarily sufficient for giving notice. Yes, you are obligated to pay rent for the time that you are an occupant of the premises. A security deposit is just that, security; it is not rent.