In theory, the following property can return that information from the Word Open XML file, using the Open XML SDK:
int pageCount = (int) document.ExtendedFilePropertiesPart.Properties.Pages.Text;
In practice, however, this isn't reliable. It might work, but then again, it might not - it all depends on 1) What Word managed to save in the file before it was closed and 2) what kind of editing may have been done on the closed file.
The only sure way to get a page number or a page count is to open a document in the Word application interface. Page count and number of pages is calculated dynamically, during editing, by Word. When a document is closed, this information is static and not necessarily what it will be when the document is open or printed.
See also https://github.com/OfficeDev/Open-XML-SDK/issues/22 for confirmation.
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