Check with a local attorney. In general they are correct regarding month-to-month, if you are still there on the 1st that means you will need to pay for a month. If you give notice on the first, you can move out a month later and not owe anything extra. If they try to sue you hire an attorney, though I'd guess they are saying you owe them extra money in order to intimidate you from suing for half of your rent. I don't practice in CA so don't rely on my advice in or out of court.
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I respectfully disagree without my out of state colleague.
According to the Department of Consumer Affairs, the 30 day clock starts on the day you give notice.
Tenant's notice to end a periodic tenancy
To end a periodic rental agreement (for example, a month-to-month agreement), you must give your landlord proper written notice before you move.
You must give the landlord the same amount of notice as there are days between rent payments.193 This means that if you pay rent monthly, you must give the landlord written notice at least 30 days before you move. If you pay rent every week, you must give the landlord written notice at least seven days before you move.194 This is true even if the landlord has given you a 60-day notice to end the rental agreement and you want to leave sooner (see Landlord's notice to end a periodic tenancy.)195
If your rental agreement specifies a different amount of notice (for example 10 days), you must give the landlord written notice as required by the agreement.196
To avoid later disagreements, date the notice, state the date that you intend to move, and make a copy of the notice for yourself. It's best to deliver the notice to the landlord or property manager in person, or mail it by certified mail with return receipt requested. (You can also serve the notice by one of the methods described under "Proper Service of Notices".)197
You can give the landlord notice any time during the rental period, but you must pay full rent during the period covered by the notice. For example, say you have a month-to-month rental agreement, and pay rent on the first day of each month. You could give notice any time during the month (for example, on the tenth). Then, you could leave 30 days later (on the tenth of the following month, or earlier if you chose to). But you would have to pay rent for the first 10 days of the next month whether you stay for those 10 days or move earlier. (Exception: You would not have to pay rent for the entire 10 days if you left earlier, and the landlord rented the unit to another tenant during the 10 days, and the new tenant paid rent for all or part of the 10 days.)198
The rental agreement or lease must state the name and address of the person or entity to whom you must make rent payments (see When You Rent section).If this address does not accept personal deliveries, you can mail your notice to the owner at the name and address stated in the lease or rental agreement. If you can show proof that you mailed the notice to the stated name and address (for example, a receipt for certified mail), the law assumes that the notice is receivable by the owner on the date of postmark.199
Attorney Kane has given you a good and useful answer.
Even though it is called "last month's rent," it is usually construed to be a deposit to be used to pay for damage beyond normal wear and tear after you move out. Ordinarily, you would have to pay the 15 days' rent to cover the end of the notice period, then the owner must send you a refund within 21 days, including an accounting for the application of any money in his / her hands as of the time you moved out. Your landlord is dead wrong about whether he has to prorate the rent. He cannot keep any money of yours that is not properly applied to rent or cleaning / repair.
You should schedule a pre-move-out inspection of the property and offer to make all needed repairs and do all necessary cleaning, to ensure that you get the full amount of your deposit back--or as much as possible. Take photos and videos, and get the dings in writing for your own protection.