Refer to this webpage for the locations of various LR files, paying attention to the one that says Serial Number:
http://helpx.adobe.com/lightroom/kb/preference-file-locations-lightroom-41.html
Go to the folder it says, and open the .lrreg file with a text editor, like Notepad or Wordpad, and look for the serial number that will be 6 groups of 4 digits beginning with 1160. Don’t post the serial-number here, of course, as this is a public forum.
If you have had more than one LR version on your computer, e.g. LR3 or LR4, there may be more than one serial number to determine, as you’ll need an older serial-number if your LR5 one is an upgrade.