No. It's should be prorated. Don't pay the full amount unless the landlord plans on always receiving the rent on the 22nd. I moved in on the 14th and instead of prorating it my landlord always collects on the 14th. If they want the rent on the first they can't charge a full month if you aren't living there. They can only charge you for the days you live there.
I do not think if that is right unless the money you have paid is for the deposit. I believe there was an agreement between the landlord and you and it should be indicated. If not, you have the right to ask the landlord about it.