From Wikipedia on 24:
“ is the only number whose divisors, namely , are exactly those numbers for which every invertible element of the commutative ring is a square root of . It follows that the multiplicative group is isomorphic to the additive group . This fact plays a role in monstrous moonshine.”
Where did that come from?
In the original “Monstrous Moonshine” paper by John Conway and Simon Norton, section 3 starts with:
“It is a curious fact that the divisors of are precisely those numbers for which implies .”
and a bit further they even call this fact:
“our ‘defining property of '”.

The proof is pretty straightforward.
We want all such that every unit in has order two.
By the Chinese remainder theorem we only have to check this for prime powers dividing .
is a unit of order in .
is a unit of order in .
A generator of the cyclic group is a unit of order in , for any prime number .
This only leaves those dividing .
But, what does it have to do with monstrous moonshine?
Moonshine assigns to elements of the Monster group a specific subgroup of containing a cofinite congruence subgroup
for some natural number where is the order of the monster-element, divides and … is a divisor of .
To begin to understand how the defining property of is relevant in this, take any strictly positive rational number and any pair of coprime natural numbers and associate to the matrix
We say that fixes if we have that
For those in the know, stands for the -dimensional integral lattice
and the condition tells that preserves this lattice under base-change (right-multiplication).
In “Understanding groups like ” Conway describes the groups appearing in monstrous moonshine as preserving specific finite sets of these lattices.
For this, it is crucial to determine all fixed by .
so we must have that is a natural number, or that is a number-like lattice, in Conway-speak.
so divides , divides and divides . As and are coprime it follows that must divide .
Now, for an arbitrary element of we have
and using our divisibility requirements it follows that this matrix belongs to if is divisible by , that is if .
We know that and that divides , so , which implies if satisfies the defining property of , that is, if divides .
Concluding, preserves exactly those lattices for which
A first step towards figuring out the Moonshine Picture.