What's the minimum number of ip addresses that a router must have?

The minimum number is of course 2, You can have at least as much IP addresses as the number of interfaces... physically. I say physically because you can have logical interfaces called subinterfaces, and even loopback interfaces. And all those interfaces can be assigned IP addresses. A physical interface can also have 2 IP addresses : one primary and one secondary, in cases where the interface is attached to a LAN with 2 subnet ranges. Subinterfaces are like f0/0.1 , f0/1.20 , ... They are specially used when we have many VLANs and want them to be able to communicate. Then, the feature called Router On A Stick is configured and it requires one subinterface per VLAN involved, so that the IP address of each subinterface can act as the default gateway of a given VLAN.

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