If, as your first post seems to suggest, the page # is in a textbox or frame, all you really need to do is click on the border of the frame of the object holding the page #, then press 'Delete'.

That said, the issue has nothing to do with compatibility between Word 2003 and Word 2010; just a lack of understanding about the different ways features such as page numbers might be inserted. The "Remove Page Numbers" function only works with page #s inserted via the "Insert Page Numbers" function during current editing session for the document; close & re-open it and Word won't remember that it's been inserted and, so, can't remove it that way.

As for Office 2016, you're better off staying with Office 2010 (IMHO) unless Office 2016 provides some added functionality you really need.

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Cheers,
Paul Edstein
[Fmr MS MVP - Word]