A month to month tenant can terminate his lease by giving a notice at least twenty days before the end of the month. RCW 59.18.200. If you held over beyond that date, you are liable for the damages--possibly an entire month's rent. If your notice to vacate was for a date mid-month, it is not effective until the end of the month and you are liable for the entire month's rent.
I am in full agreement with Mr. Cutting's answer. However, I might suggest you contact your landlord and discuss the situation. Sometimes a landlord may be willing to pro-rate your move-out if you have had a good relationship with the landlord and certain conditions exist, such as leaving the premises move-in ready. It never hurts to ask.
The foregoing is offered as general information only, and does not constitute legal advice. Please be advised you should consult a lawyer for advice specific to your situation.The foregoing should not be construed to establish an attorney client relationship.
I agree with Mr. Cutting's answer, and I would add that if you did not give notice more than 20 days before the end of the prior month, you would be liable for the following month's rent even if you vacated on the last day of the month. If your notice was given in time for vacating at the end of the month, but you stayed until the 6th of the following month, your notice was effectively revoked and/or you held over after termination of the rental term - either way you were liable for the full month's rent. The bottom line is that a landlord has no obligation to prorate the last month's rent, and has no obligation to terminate a month-to-month rental agreement on a date other than the last of the month, unless the agreement is in writing and specifies that each month is something other than the calendar month (e.g., from the 6th to the 6th). I have never seen a month-to-month rental agreement written that way.
The foregoing is not intended as legal advice, but solely as general information. No attorney/client relationship exists between the poster and the answering attorney.