Upon receipt of an application for registration of any aircraft and payment of the required fee, the division shall file the application, assign to the application and the aircraft to be registered a distinct number and register the application and the number in an index to be kept for that purpose. A number plate and certificate of registration bearing the same number assigned to the application shall be delivered to the owner or lessee of the aircraft without further expense to such applicant. The certificate of registration shall be carried in the aircraft at all times and the registration number shall be conspicuously displayed on the aircraft as may be required by appropriate rule of the director.