I have 2 301's and they do have a number on the bottom of the machine...I think they all have 2 letters and 6 digits too...on the same side as the extension bed..right underneath where the fold is..
$10 is a fantastic price..never pass one of these machines up ..just the bobbin case and the needle plate are worth maybe up to $100 because they fit on a 221 machine...and sometimes the foot pedal too...not the attachments though