24 (twenty-four) is the natural number following

23 and preceding 25.

24 is the factorial of 4 and a composite number,

being the first number of the form 23q, where q

is an odd prime

It is the smallest number with exactly eight divisors:

1, 2, 3, 4, 6, 8, 12, and 24. It is a highly composite

number, having more divisors than any smaller

number. Adding up all the proper divisors of 24

except 4 and 8 gives 24, hence 24 is a semiperfect

number.

Subtracting one from any of its divisors (except

1 and 2, but including itself) yields a prime number.

24 is the largest number with this property, for to

have this property a number cannot be divisible

by a prime greater than three, nor can it be divisible

by 9 or 16.

24 has an aliquot sum of 36 and the aliquot sequence

(24,36,55,17,1,0). Twenty-four is only the aliquot sum

of one number the square 529.

There are 10 solutions to the equation φ(x) = 24, namely

35, 39, 45, 52, 56, 70, 72, 78, 84 and 90. This is more

than any integer below 24, making 24 a highly totient number.

24 is a nonagonal number. This number is also the sum

of a twin prime (11 + 13). It is a Harshad number and

a semi-meandric number.

Together with the number one, 24 is one of the few

numbers n for which the sum of μ(d)d2 over the

divisors d of n is equal to itself.

The product of any four consecutive numbers

is divisible by 24. This is because, among any

four consecutive numbers, there must be two

even numbers, one of which is a multiple of

four, and there must be a multiple of three.

In 24 dimensions there are 24 even positive definite

unimodular lattices, called the Niemeier lattices.

One of these is the exceptional Leech lattice which

has many surprising properties; due to its existence,

the answers to many problems such as the kissing

number problem and sphere packing are known in

24 dimensions but not in many lower dimensions.

The Leech lattice is closely related to the equally

nice length-24 binary Golay code and the Steiner

system S(5,8,24) and the Mathieu group M24.

One construction of the Leech lattice is possible

because of the remarkable fact that

12+22+32+...+242 =702 is a perfect square;

24 is the only integer greater than 1 with this

property. These properties of 24 are related

to the fact that the number 24 also appears

in several places in the theory of modular

forms; for example, the modular discriminant

is the 24th power of the Dedekind eta function.

The Barnes-Wall lattice contains 24 lattices.

24 is the highest number n with the property

that every element of the group of units

(Z/nZ)* of the commutative ring Z/nZ, apart

from the identity element, has order 2; thus

the multiplicative group

(Z/24Z)* = {1,5,7,11,13,17,19,23} is isomorphic

to the additive group (Z/2Z)3. This fact plays a

role in monstrous moonshine.

The 24-cell, with 24 octahedral cells and 24

vertices, is a self-dual convex regular 4-polytope;

it has no good 3-dimensional analogue.

For any prime p greater than

3, p2 − 1 is divisible by 24 with no remainder.

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