It is very important that these be linkeble. Can they be? Otherwise I would need to use 4 remotes?

One remote can be used for multiple sets. If the receivers are not next to each other you would just have to walk around with the remote so you are in close enough range for the receiver to pick up the signal from the remote. I have 4 sets in my kitchen and use 1 remote. The upside is each segment can be controlled independently, for example I can make the above cabinet lights blue for mood, while keeping the under cabinet lights warm white for task lighting.

You can link them together with strip tape light jumper connectors and use one remote.

No, these are not linkable.