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From Wikipedia, the free encyclopedia

← 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
Hexadecimal1816

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

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Transcription

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 exactly eight divisors: 1, 2, 3, 4, 6, 8, 12 and 24;[1] thus, it is a highly composite number, having more divisors than any smaller number.[2] 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.[3]
  • 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.[4]
  • 24 is a refactorable number, as it has exactly eight positive divisors, and 8 is one of them.
  • 24 is a twin-prime sum, specifically the sum of the third pair of twin primes .
  • 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.[5]
  • 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.[6]
  • 24's digits in decimal can be manipulated to form two of its factors, as 2 * 4 is 8 and 2 + 4 is 6. In turn 6 * 8 is 48, which is twice 24, and 4 + 8 is 12, which is half 24.
  • 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.[7]
  • 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).[8]
  • 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 geometry

In science

In religion

In music

In sports

In other fields

Astronomical clock in Prague

24 is also:

References

  1. ^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-06.
  2. ^ "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  3. ^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  4. ^ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  5. ^ "Sloane's A097942 : Highly totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  6. ^ "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-10.
  8. ^ Weisstein, Eric W. "Cannonball Problem". mathworld.wolfram.com. Retrieved 2020-08-19.
  9. ^ Coxeter, H.S.M. (1991), Regular Complex Polytopes (2nd ed.), Cambridge: Cambridge University Press
  10. ^ 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. hdl:11858/00-001M-0000-0029-C3D7-E. ISSN 0033-4545. S2CID 101719914.
  11. ^ "Revelation 4:4, New International Version (1984)". Bible.cc. Retrieved 2013-05-03.
  12. ^ "Is 24K gold pure?". Scientific American. Retrieved 2020-08-12.
  13. ^ "Greek alphabet | History, Definition, & Facts". Encyclopedia Britannica. Retrieved 2020-08-12.
  14. ^ "GammonSite - Rules of backgammon". www.gammonsite.com. Retrieved 2020-08-12.

External links

This page was last edited on 18 March 2024, at 15:25
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