It is called 統一編號 according with http://www.foxsemicon.com/chinese/index.php and other sources. In English it is called "Unified Business No." or "Unified Business Number", being "UBN" the acronym.
Great explanation, but there are some actual valid numbers this simplified algorithm does not catch, so we will be using a more complex version outlined in some of the other implementations below.
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Algorithm:
Examples: