The modem has to be authorized via MAC address on the network. They know the MACs of the modems they own, so it's very simple to know if a modem is theirs or not.
There is no way for them to differentiate whether the device behind the modem is a router or a computer.
>You must maybe use their modem (how they can check it I don't know, maybe they don't give you the configuration parameters for the network?)
Because I answered the "how?"