The tags gcd is already synonym of divisibility.
It seems logical that the least-common-multiple should also be a synonym. (And maybe also lcm could be created and added to the synonyms; so that some user does not create this tag as a separate tag in the future.) As I do not have sufficiently high score in divisbility to suggest a synonym, I am posting this suggestion here.
The recent revisions of (least-common-multiple) tag-excerpt and tag-wiki specifically mention that the notion of lcm exists not only for integers but also in rings. I don't think this should cause a problem, since rings are also mentioned in the tag-wiki for divisibility. (Although it not explicitly mentioned at the current version of tag-excerpt.) Note that the tag-wiki for divisibility explicitly mentions both gcd and lcm.