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The dropbox software running on your work computer notices that the file has been modified, and uploads it to the dropbox servers.
Then on your home computer, the dropbox software notices that the modification date of the version on the dropbox servers is more recent than the modification date of the file on the home computer, and so downloads it from dropbox and replaces the older version.
And vice-versa, of course, if you modify the file on your home computer.
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