# 24 (number)

(Redirected from Twenty-four)
 ← 23 24 25 →
Cardinaltwenty-four
Ordinal24th
(twenty-fourth)
Numeral systemtetravigesimal
Factorization23 × 3
Divisors1, 2, 3, 4, 6, 8, 12, 24
Greek numeralΚΔ´
Roman numeralXXIV
Binary110002
Ternary2203
Senary406
Octal308
Duodecimal2012

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

## 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 2qq, where q is an odd prime. It is the smallest number with at least 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.[1] 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 = 5/2 × 24
• 24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24.[2]
• 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.[3]
• 24 is a refactorable number, as it has exactly eight positive divisors, and 8 is one of them.
• 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.[4]
• 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.[5]
• 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.
• 24 is the number of digits of the fifth and largest known unitary perfect number, when written in decimal: 146361946186458562560000.[6]
• 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).[7]
• 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/nZ is a square root of 1. It follows that the multiplicative group of invertible elements (Z/24Z)× = {±1, ±5, ±7, ±11} 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 6K ± 1), and in particular any prime n greater than 3, has the property that n2 – 1 is divisible by 24.
• The modular discriminant Δ(τ) is proportional to the 24th power of the Dedekind eta function η(τ): Δ(τ) = (2π)12η(τ)24.

## In other fields

Astronomical clock in Prague

24 is also:

• The number of bits a computer needs to represent 24-bit color images (for a maximum of 16,777,216 colours—but greater numbers of bits provide more accurate colors).
• The number of karats representing 100% pure gold.[11]
• The number of cycles in the Chinese solar year.
• The number of years from the start of the Cold War until the signing of the Seabed Arms Control Treaty, which banned the placing of nuclear weapons on the ocean floor within certain coastal distances.
• The number of frames per second at which motion picture film is usually projected, as this is sufficient to allow for persistence of vision.
• The number of letters in both the modern and classical Greek alphabet.[12] 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.[13]
• 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 24 Game).
• 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".

## References

1. ^ "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
2. ^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
3. ^ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
4. ^ "Sloane's A097942 : Highly totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
5. ^ "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
6. ^ Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
7. ^ Weisstein, Eric W. "Cannonball Problem". mathworld.wolfram.com. Retrieved 2020-08-19.
8. ^ Coxeter, H.S.M. (1991), Regular Complex Polytopes (2nd ed.), Cambridge: Cambridge University Press
9. ^ Meija, Juris; Coplen, Tyler B.; Berglund, Michael; Brand, Willi A.; Bièvre, Paul De; Gröning, Manfred; Holden, Norman E.; Irrgeher, Johanna; Loss, Robert D.; Walczyk, Thomas; Prohaska, Thomas (2016-03-01). "Atomic weights of the elements 2013 (IUPAC Technical Report)". Pure and Applied Chemistry. 88 (3): 265–291. doi:10.1515/pac-2015-0305. ISSN 0033-4545. S2CID 101719914.
10. ^ "Revelation 4:4, New International Version (1984)". Bible.cc. Retrieved 2013-05-03.
11. ^ "Is 24K gold pure?". Scientific American. Retrieved 2020-08-12.
12. ^ "Greek alphabet | History, Definition, & Facts". Encyclopedia Britannica. Retrieved 2020-08-12.
13. ^ "GammonSite - Rules of backgammon". www.gammonsite.com. Retrieved 2020-08-12.