Is the product key for Microsoft office 2013 the same as all product keys for 2013 Microsoft offices that operate under laptops using windows 8.1?

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The product key is associated with the MS Office 2013 software, not the specific MS Windows version of Operating System - unless the MS Office lists a minimum version of OS ie: "Windows 8.0" In this case, it would not install on Windows 7. Otherwise, you should be able to install it on Windows 7, Windows 8, or later.