Consistent hashing.

The divisor is the trap: every key’s home is computed from N, so changing N re-homes nearly all of them at once.

A server’s position comes from hashing its name — nothing about it depends on how many others exist.

Each key walks clockwise to the first server it meets; no arithmetic ever mentions the server count.

The newcomer inherits only the arc just behind it — keys on the rest of the circle never hear the news.

Departure is the mirror image: the orphaned arc slides to the next server along, and nobody else moves.

Entering each server many times under different names irons out the luck of where the hashes landed.