Take the 2-minute tour ×  In What's so special about the
divisors of $24$? it is noted that the exponent of the group of
units modulo $n$, that is the highest order of an element of
$U(n):=(\Bbb Z/n\Bbb Z)^\times$, is precisely $2$ if and only if
the integer $n$ divides $24$. An elementary argument is given,
as well as some analytic machinery, but I recall (perhaps not
quite accurately) that the number $24$ shows up a lot in high-
powered math related to number theory, like lattices, moonshine,
modular forms, string theory etc: suspicious.  If I were a
believer in the magical and skeptical of coincidences, I might
want to know if there is a high-powered explanation of this fact
from the cited theoretical areas (not including asymptotic or
statistical heuristics from analytic number theory). Or is it
merely a collision of small numbers?  share|improve this
question I think the explanation might go in the other
direction: Gannon in Moonshine beyond the Monster speculates
that this property of $24$ is responsible