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Computer and bitcoin addresses are not tied together other than that you install and run bitcoin client on it, which is basically bitcoin wallet application. In there you can have many addresses.
You can move wallet file that bitcoin client setup on your computer somewhere else for safe keeping for instance and still receive bitcoins sent to those addresses that you generated within bitcoin client.
First time you install and run bitcoin client it will take some time, could be few hours to get all blocks from network's blockchain, once all blocks downloaded then you can use the client, generate address and get bitcoins sent to it.
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