I have a machine with ArcGIS for Desktop 10.0 installed on it.
I need to find out which license it's using.
The machine is single use ArcEditor and was authorized online.
How can I find what the license number is?
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go to customers.esri.com and log in as the administrator for your company.
go to authorization and provisioning.
on the reporting tab you will see each auth and de-auth transaction.
you can compare the authorization date with the date authorized in the arcgis administrator mentioned by Aaron.
Dont forget you can (should) de-authorize a machine that is not being used any longer (or before you move that license to another machine)
If you still can't tell which license it is you will need to call customer service with the aforementioned fulfilment ID.
Go to the ArcGIS Administrator located on the Start Menu > All Programs > ArcGIS > ArcGIS Administrator. From there look at the "License Manager Environment Variable Information" section under "ARCGIS_LICENSE_FILE:xxxxxx". Also look at the "Machine IDs..." tab on the main layout.
Edit (Additional information from @Brad):
Using the ArcGIS Administrator, select the desired product folder (Desktop or Engine) in the table of contents.
Select the desired single use product.
Select Availability to view the licenses available for use on the local machine. This table indicates whether the feature has been installed and authorized. The license expiration date is also provided.
For single use features, double-click the feature to view its fulfilment ID. The fulfilment ID can be used to track down the particular authorization transaction when troubleshooting authorization related problems.