# A high-powered explanation for $\exp U(n)=2\iff n\mid24$?

In What's so special about the divisors of $$24$$? (Math. Mag., 2012) 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 (see also answers here for a one-line proof), 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?

• I think the explanation might go in the other direction: Gannon in Moonshine beyond the Monster speculates that this property of $24$ is responsible for its appearance in high-powered math. Oct 19, 2012 at 18:06
• Wikipedia echoes this, stating "This fact plays a role in monstrous moonshine."
– anon
Oct 19, 2012 at 18:09
• $\exp U(n) = \lambda(n)$, where $\lambda$ is the Carmichael function. From that follows an easy explanation.
– lhf
Oct 19, 2012 at 18:24

$\qquad$