Hola David -
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Yes, in principle, airplanes from Boeing take the "customer" designation number of the client airline to which first delivered, and therafter, keep that designation, even sold later to another airline... Realize that each customer airline have their "options", such as type of engine, or configuration. So, for me, with Boeing airplanes, a customer numbers tells me much about what airplane it is, and what it normally has, as equipment.
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Has happened that airplanes were ordered by "ABC" airline, then politics changed, and the USA prohibited export of these airplanes to that airline, but delivered such airplanes to "XYZ" airline, at times with the original "ABC" airline designation, at times with the "XYZ" designation. In the 747, was the case of airplanes ordered originally by Iran Air, but delivered to Saudia, or airplanes ordered by Libyan Arab, but delivered to Varig...
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There are other rare exceptions. It has happened that Boeing airplanes are sold to a leasing company (which also have a Boeing customer designation), then immediately leased to an airline. Has been the case of some 737s... which I am not too familiar with.
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Airbus does not use that system... Somebody else will tell you.
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Happy contrails