It entirely depends on the lease you signed when you moved in. Read your lease to find out. Generally, most leases charge an initial late fee after a certain number of days of delinquency, usually three to five. On top of that, they charge a daily fee for each day the rent is late, which you could call "interest" if you wish. In any event, the controlling document is your lease and it should spell out the specifics of delinquency penalties.
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