Apple ID
How do I add a Mac to my devices on my Apple ID?
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iTunes Store- Associating a device or computer to your Apple ID
Your Apple ID is Limited to 10 Devices
If all you wish to do is authorize your computer:
Authorization
iTunes Store - Authorize or deauthorize your Mac or PC.
In iTunes you use the Authorize This Computer or De-authorize This Computer option under the Store menu in iTunes' menubar. For Windows use the ALT-S keys to access it. Or turn on Windows 7 and 8 iTunes menus: iTunes- Turning on iTunes menus in Windows 8 and 7.