Do you have to pay lasts months rent if you paid secruity deposit?

The security deposit is a deposit against possible damage to the property.  It is not rent.  Generally, you are required to pay the last month's rent and provide the landlord with a forwarding address to return your security deposit to you.

It sounds like you want to know whether you should pay this last month's rent because you are concerned that the Landlord has your money. You have a right to be concerned. While legally you are responsible for the rent, practically speaking many tenants put their security towards the last month's rent. A security deposit can serve a dual purpose, one for damages upon moving out and two as security against any unpaid rent. Many Landlords are fine with applying the security for the last month. You and the Landlord are free to agree to whatever you want, so find out if that is agreeable to her.