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1) What state are you in?
2) Is this a month-to-month rental agreement?
3) If so, from when to when is the rent paid (for eg. from the first to the last day of the month, from the 15th of the month to the 14th of the next month, etc.)?
A tenant who pays the rent monthly must give the landlord a full 30 days notice before leaving. If the tenant leaves before the 30 days, he remains liable for the rent for the entire 30 days, unless the landlord rents the unit to someone else within the remainder of the 30 days.
The 30 days starts to run on the day after the day your receive the notice (the actual day you receive the notice is not included in the 30 days) - not the date written on the notice.
Given the information you provided, your ex-tenant is responsible for the rent until September 6. It does not matter that the day you received the notice was a Saturday.
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If the new tenant moves in after September 5, your ex-tenant remains responsible for the rent until September 5.