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Photoshop CLOUD does not use a serial number... what version do you have?
As John mentioned above, the Creative Cloud software does not use serial #'s. If you are being prompted to enter a serial #. Please see this help doc. http://helpx.adobe.com/creative-cloud/kb/ccm-prompt-serial-number.html
If you are using a previous CS6 and earlier Photoshop version. Where did you purchase the software? If you have purchased the software from Adobe or an authorized reseller
Then I would suggest to make sure that your are logged in as an administrator and go thru the steps in this help doc. Check the hosts file to ensure that you can activate the software on your computer.
If you are still having issues, you can contact our chat agents here to check the serial # to ensure that it is valid