I am in Massachusettes. I have given a 30 day plus 3 days of the next month notice. I am an at will tenant. The landlord is telling me that I need to pay a full months rent for the extra 30 days is this right or is it prorated? They are also telling me that I have to prove I paid my last months rent when I signed a one year lease 5 years ago, I did not ever receive a copy of the lease however I do have a copy of a rent increase letter from 2008 that informs me to also bring my last months rent up to the increase is this good enough because I do not have the cancelled check?

Yes, if the landlord says you did not pay the last month rent when you moved in, you will have to prove it. If you can't prove it, if he sues you how will you prove it in court?

They do not have to prorate anything unles that is in a written rental agreement or lease.

So if you are going to say one day into the next month, you could be billed for that entire month.

This is normal rental for business that are in rental.