The correct answer is A) official number. When offloading garbage to another ship, the records must identify that ship by its official number. This is a requirement specified in the regulations governing the handling and disposal of garbage from ships, as outlined in Annex V of the International Convention for the Prevention of Pollution from Ships (MARPOL). The official number is a unique identifier assigned to each vessel and is typically displayed on the ship's hull or documentation, which allows for proper tracking and verification of the garbage disposal process. The other options, such as the ship's home port (B), the master's name (C), or the operator's name of record (D), while potentially useful information, are not the specific requirement for identifying the receiving vessel during garbage offloading operations.