I know one of the uses -
if you're a teacher who works with a class with 27 pupils and a class
with 18 pupils, and you want each class to have a sports tournament with
the winning team from each class playing each other, you would want to
know the largest possible team number that you could break both classes
into. In this case, both classes can be split into teams with 9 players,
as that's the HCF.