# 24 (number)

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

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Cardinal | twenty-four | |||

Ordinal | 24th (twenty-fourth) | |||

Numeral system | tetravigesimal | |||

Factorization | 2^{3}× 3 | |||

Divisors | 1, 2, 3, 4, 6, 8, 12, 24 | |||

Greek numeral | ΚΔ´ | |||

Roman numeral | XXIV | |||

Binary | 11000_{2} | |||

Ternary | 220_{3} | |||

Quaternary | 120_{4} | |||

Quinary | 44_{5} | |||

Senary | 40_{6} | |||

Octal | 30_{8} | |||

Duodecimal | 20_{12} | |||

Hexadecimal | 18_{16} | |||

Vigesimal | 14_{20} | |||

Base 36 | O_{36} |

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

## Contents

## In mathematicsEdit

- 24 is the factorial of 4 (24 = 4!) and a composite number, being the first number of the form 2
^{3}*q*, where*q*is an odd prime. - Since 24 = 4!, it follows that 24 is 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).
- 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.
^{[1]} - 24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24.
^{[2]} - 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.
- 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.^{[3]} - 24 is a nonagonal number.
^{[4]} - 24 is the sum of the prime twins 11 and 13.
- 24 is a Harshad number.
^{[5]} - 24 is a semi-meandric number.
- 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.
- The tesseract has 24 two-dimensional faces (which are all squares).
- 24 is the only nontrivial solution to the cannonball problem, that is: 1
^{2}+ 2^{2}+ 3^{2}+ … + 24^{2}is a perfect square (70^{2}). (The trivial case is just 1^{2}= 1^{2}.) - 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 1^{2}+ 2^{2}+ 3^{2}+ ... + 24^{2}= 70^{2}.) - The modular discriminant Δ(
*τ*) is proportional to the 24th power of the Dedekind eta function*η*(*τ*): Δ(*τ*) = (2π)^{12}*η*(*τ*)^{24}. - The Barnes–Wall lattice contains 24 lattices.
- 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**/24**Z**)^{×}= {±1, ±5, ±7, ±11} is isomorphic to the additive group (**Z**/2**Z**)^{3}. This fact plays a role in monstrous moonshine.- It follows that any number
*n*relatively prime to 24, and in particular any prime*n*greater than 3, has the property that*n*^{2}– 1 is divisible by 24.

- It follows that any number
- The 24-cell, with 24 octahedral cells and 24 vertices, is a self-dual convex regular 4-polytope of rotational symmetry order 576 (equal to 24 squared). It tiles 4-dimensional space in a 24-cell honeycomb, within which twenty-four 24-cells surround a single 24-cell.
- 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.)
- 24 is the largest integer that is divisible by all natural numbers no larger than its square root.
- 24 is the Euler characteristic of a K3 surface
- 24 is the smallest 5-hemiperfect number
- 24 is the order of the cyclic group equal to the stable 3-stem in homotopy groups of spheres: π
_{n+3}(*S*^{n}) =**Z**/24**Z**for all*n*≥ 5.

## In scienceEdit

- The atomic number of chromium
- The number of hours in a day
- 24! is an approximation (exceeding by just over 3%) of the Avogadro constant.

## In religionEdit

- The number of books in the Tanakh.
- In Christian apocalyptic literature it represents the complete Church, being the sum of the 12 tribes of Israel and the 12 Apostles of the Lamb of God. For example, in
*The Book of Revelation*: "Surrounding the throne were twenty-four other thrones, and seated on them were twenty-four elders. They were dressed in white and had crowns of gold on their heads."^{[6]} - Number of Tirthankaras.
- Number of spokes in the Ashok Chakra.

## In musicEdit

- There are a total of 24 major and minor keys in Western tonal music, not counting enharmonic equivalents. Therefore, for collections of pieces written in each key, the number of pieces in such a collection; e.g., Chopin's
*24 Preludes*.

## In sportsEdit

- Four-and-Twenty was an American racehorse.
- In association football:
- The FIFA World Cup final tournament featured 24 men's national teams from 1982 to 1994.
- The FIFA Women's World Cup final tournament featured 24 national teams in 2015 and 2019.

## In other fieldsEdit

**24** is also:

- The number of bits a computer needs to represent TrueColor images (for a maximum of 16,777,216 colours). (But greater numbers of bits provide more accurate colors. "TrueColor" is one of many possible representations of colors.)
- The number of carats representing 100% pure gold.
- The number of cycles in the Chinese solar year.
- The number of frames per second at which motion picture film is usually projected.
- The number of letters in both the modern and classical Greek alphabet. For the latter reason, also the number of chapters or "books" into which Homer's
*Odyssey*and*Iliad*came to be divided. - The number of runes in the Elder Futhark
- The number of points on a backgammon board.
- A children's mathematical game involving the use of any of the four standard operations on four numbers on a card to get 24 (see Math 24)
- The maximum number of Knight Companions in the Order of the Garter
- The number of the French department Dordogne.
- Four and twenty is the number of blackbirds baked in a pie in the traditional English nursery rhyme
*Sing a Song of Sixpence*.

## ReferencesEdit

**^**"Sloane's A002182 : Highly composite numbers".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-05-31.**^**"Sloane's A005835 : Pseudoperfect (or semiperfect) numbers".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-05-31.**^**"Sloane's A097942 : Highly totient numbers".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-05-31.**^**"Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-05-31.**^**"Sloane's A005349 : Niven (or Harshad) numbers".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-05-31.**^**"Revelation 4:4, New International Version (1984)". Bible.cc. Retrieved 2013-05-03.

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