Instructions
Turn on the Xbox 360 that does not yet have your gamertag installed.
Press the ̶0;Xbox Guide̶1; button on your controller then press the ̶0;X̶1; button.
Select ̶0;Yes̶1; when you are prompted to sign out of your account.
Press the ̶0;Xbox Guide̶1; button a second time and choose ̶0;Recover Gamertag̶1; from the menu.
Confirm that you want to recover your gamertag on that console.
Provide your Windows Live ID and password when you are prompted.
Allow the Xbox 360 to recover your gamertag based on the Windows Live ID information that you provided. It may take a little time to transfer the account information. Once the account has been transferred, you will be able to log into the Xbox Live community using your gamertag.