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The computers in my office share the same IP address. Having that in mind, how does the Internet know that when I request a file from a remote server, it is exactly my computer that should receive the file and not one of the other computers (in my office) that share the same IP address?

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"All the computers in my office use the same IP address." No they don't. You can't have two machines on the same segment with the same IP address without a lot of trickery. – Ignacio Vazquez-Abrams Nov 15 '11 at 10:06
I'm pretty sure he meant "the same IP address" as viewed from the Internet point of view. – Antoine Benkemoun Nov 15 '11 at 10:08
@Antoie, That's right. – Emanuil Rusev Nov 15 '11 at 10:09
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It uses the tuple (source ip, source port, dest port, dest ip).

Therefore it does not identify a computer but a connection and that is all that matters from an Internet-server point of view.

You router then has a table that allows him to know to which computer he needs to forward the connection. This is called NAT.

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