XXIV Corps Artillery

24 (twenty-four) is the natural number following 23 and preceding 25.

The SI prefix for 10²⁴ is yotta (Y), and for 10⁻²⁴ (i.e., the reciprocal of 10²⁴) yocto (y). These numbers are the largest and smallest number to receive an SI prefix to date.

In mathematics

24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2''q'', where ''q'' is an odd prime. It is the smallest number with exactly eight positive divisors: 1, 2, 3, 4, 6, 8, 12, and 24; thus, it is a highly composite number, having more divisors than any smaller number. Furthermore, it is an abundant number, since the sum of its proper divisors ( 36) is greater than itself, as well as a superabundant number.

In number theory and algebra

*24 is the smallest 5-hemiperfect number, as it has a half-integer abundancy index:
*:1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 =  × 24
*24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24.
*24 is a practical number, since all smaller positive integers than 24 can be represented as sums of distinct divisors of 24.
*24 is a Harshad number, since it is divisible by the sum of its digits in decimal.
*24 is a highly totient number, as there are 10 solutions to the equation ''φ''(''x'') = 24, which is more than any integer below 24. 144 (the square of 12) and 576 (the square of 24) are also highly totient.
*24 is a polite number, an amenable number, an idoneal number, and a tribonacci number.
*24 forms a Ruth-Aaron pair with 25, since the sums of distinct prime factors of each are equal ( 5).
*24 is a compositorial, as it is the product of composite numbers up to 6.
*24 is a pernicious number, since its Hamming weight in its binary representation (11000) is prime (2).
*24 is the third nonagonal number.
*24 is a congruent number, as 24 is the area of a right triangle with a rational number of sides.
*24 is a semi-meandric number, where an order-6 semi-meander intersects an oriented ray in R² at 24 points.
*Subtracting 1 from any of its divisors (except 1 and 2 but including itself) yields a prime number; 24 is the largest number with this property.
*24 is the largest integer that is divisible by all natural numbers no larger than its square root.
*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 at least one multiple of three.
* 24 = 4!, the factorial of 4. It is the largest factorial that does not contain a trailing zero at the end of its digits (since factorial of any integer greater than 4 is divisible by both 2 and 5), and represents the number of ways to order 4 distinct items:
*:(1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2), (2,1,3,4), (2,1,4,3), (2,3,1,4), (2,3,4,1), (2,4,1,3), (2,4,3,1), (3,1,2,4), (3,1,4,2), (3,2,1,4), (3,2,4,1), (3,4,1,2), (3,4,2,1), (4,1,2,3), (4,1,3,2), (4,2,1,3), (4,2,3,1), (4,3,1,2), (4,3,2,1).
*24 is the only nontrivial solution to the cannonball problem; that is, 1² + 2² + 3² + … + 24² is a perfect square (70²).
*24 is the only number whose divisors — 1, 2, 3, 4, 6, 8, 12, 24 — are exactly those numbers ''n'' for which every invertible element of the commutative ring Z/''n''Z is a square root of 1. It follows that the multiplicative group of invertible elements (Z/24Z)^× = is isomorphic to the additive group (Z/2Z)³. This fact plays a role in monstrous moonshine.
*:It follows that any number ''n'' relatively prime to 24 (that is, any number of the form 6''K'' ± 1), and in particular any prime ''n'' greater than 3, has the property that ''n''² – 1 is divisible by 24.
*The modular discriminant is proportional to the 24th power of the Dedekind eta function : .

In geometry

*24 degrees is the measure of the central angle and external angle of a pentadecagon.
*An icositetragon is a regular polygon with 24 sides and Dih₂₄ symmetry of order 48. It can fill a plane-vertex alongside a triangle and octagon.
*24 is the Euler characteristic of a K3 surface: a general elliptic K3 surface has exactly 24 singular fibers.
*24 is the order of the octahedral group — the group of rotations of the regular octahedron and the group of rotations of the cube. The binary octahedral group is a subgroup of the 3-sphere ''S''³ consisting of the 24 elements of the binary tetrahedral group along with the 24 elements contained in its coset . These two cosets each form the vertices of a self-dual 24-cell, and the two 24-cells are dual to each other. (See point below on 24-cell).
*24 is the count of different elements in various uniform polyhedron solids. Within the family of Archimedean and Catalan solids, there are 24 edges in a cuboctahedron and rhombic dodecahedron, 24 vertices in a rhombicuboctahedron, truncated cube, truncated octahedron, and snub cube, as well as 24 faces in a deltoidal icositetrahedron, tetrakis hexahedron, triakis octahedron, and pentagonal icositetrahedron. The cube-octahedron compound, with a rhombic dodecahedral convex hull, is the first stellation of the cuboctahedron, with a total of 24 edges.
*:There are 12 non-prismatic uniform polyhedron compounds ( UC₀₁, UC₀₃, UC₀₈, UC₁₀, UC₁₂, UC₃₀, UC₄₂, UC₄₆, UC₄₈, UC₅₀, UC₅₂, and UC₅₄) and 12 uniform star polyhedra ( U₀₃, U₁₃, U₁₄, U₁₅, U₁₇, U₁₈, U₁₉, U₂₁, U₃₆, U₃₇, U₄₁, and U₅₈) with a vertex, edge, or face count of 24. The great disnub dirhombidodecahedron, also called ''Skilling's figure,'' is a degenerate uniform star polyhedron with a Euler characteristic of 24, when pairs of coinciding edges are considered to be single edges.
*:Finally, 6 Johnson solids ( J₁₇, J₂₇, J₃₇, J₄₅, J₆₁, and J₉₀) also have vertex, edge, or face counts of 24. The pseudo great rhombicuboctahedron, one of two known pseudo-uniform polyhedra alongside the elongated square gyrobicupola (J₃₇), has 24 vertices.
*The tesseract has 24 two-dimensional faces (which are all squares). Its dual four-dimensional polytope is the 16-cell, which has 24 edges.
*The 24-cell, with 24 octahedral cells and 24 vertices, is a self-dual convex regular 4-polytope. It possesses 576 (24×24) rotational symmetries and 1152 isometries altogether. It tiles 4-dimensional space in a 24-cell honeycomb, in which each 24-cell is surrounded by 24 24-cells.
*:The vertices of the 24-cell honeycomb can be chosen so that in 4-dimensional space, identified with the ring of quaternions, they are precisely the elements of the subring generated by the binary tetrahedral group as represented by the set of 24 quaternions $\$ in the D₄ lattice. Known as the ring of Hurwitz integral quaternions, this set of 24 quaternions forms the set of vertices of a single 24-cell, all lying on the sphere ''S''³ of radius one centered as the origin. ''S''³ is the Lie group ''Sp(1)'' of unit quaternions (isomorphic to the Lie groups ''SU(2)'' and ''Spin(3)''), and so the binary tetrahedral group — of order 24 — is a subgroup of ''S''³.
*:The 24 vertices of the 24-cell are contained in the regular complex polygon ₄₄, or of symmetry order 1152, as well as 24 4-edges of 24 octahedral cells (of 48). Its representation in the F₄ Coxeter plane contains two rings of 12 vertices each.
*: Truncations, runcinations, and omnitruncations of the 24-cell yield polychora whose Petrie polygons are 24-sided icositetragons; i.e., within the truncated 24-cell, runcinated 24-cell, and omnitruncated 24-cell, amongst others.
*24 is the kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 24 such spheres form the vertices of a 24-cell).
*The Barnes–Wall lattice contains 24 lattices.
*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 densest lattice sphere-packing problem 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 ''M''₂₄. (One construction of the Leech lattice is possible because 1² + 2² + 3² + ... + 24² = 70²).
*24 is the order of the cyclic group equal to the stable 3-stem in homotopy groups of spheres: _''n''+3(''S''^''n'') = Z/24Z for all ''n'' ≥ 5.

In science

* The atomic number of chromium.
* The average number of hours in a day (on Earth), also known as a mean solar day.
* 24! is an approximation (exceeding by just over 3%) of the Avogadro constant.

In religion

*The number of books in the Tanakh

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