21 (number)

For other uses, see 21 (disambiguation).

← 20 21 22 →
← 20 21 22 23 24 25 26 27 28 29 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	twenty-one
Ordinal	21st (twenty-first)
Factorization	3 × 7
Divisors	1, 3, 7, 21
Greek numeral	ΚΑ´
Roman numeral	XXI
Binary	101012
Ternary	2103
Senary	336
Octal	258
Duodecimal	1912
Hexadecimal	1516

21 (twenty-one) is the natural number following 20 and preceding 22.

In mathematics

21 is:

• a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.
• a Fibonacci number.
• a Harshad number.
• a Motzkin number.
• a triangular number.
• an octagonal number.
• a composite number, its proper divisors being 1, 3 and 7.
• the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21), making it a triangular number.
• the sum of the sum of the divisors of the first 5 positive integers.
• the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number.
• a repdigit in base 4 (111₄).
• the smallest natural number that is not close to a power of 2, 2ⁿ, where the range of closeness is ±n.
• the smallest number of differently sized squares needed to square the square.
• the largest n with this property: for any positive integers a,b such that a + b = n, at least one of ${\tfrac {a}{b}}$ and ${\tfrac {b}{a}}$ is a terminating decimal. See a brief proof below.

Note that a necessary condition for n is that for any a coprime to n, a and n - a must satisfy the condition above, therefore at least one of a and n - a must only have factor 2 and 5. Let $A(n)$ donate the quantity of the numbers smaller than n that only have factor 2 and 5 and that are coprime to n, we instantly have ${\frac {\varphi (n)}{2}}<A(n)$. We can easily see that for sufficiently large n, $A(n)\sim {\frac {\log _{2}(n)\log _{5}(n)}{2}}={\frac {\ln ^{2}(n)}{2\ln(2)\ln(5)}}$, but $\varphi (n)\sim {\frac {n}{e^{\gamma }\;\ln \ln n}}$, $A(n)=o(\varphi (n))$ as n goes to infinity, thus ${\frac {\varphi (n)}{2}}<A(n)$ fails to hold for sufficiently large n. In fact, For every n > 2, we have $A(n)<1+\log _{2}(n)+{\frac {3\log _{5}(n)}{2}}+{\frac {\log _{2}(n)\log _{5}(n)}{2}}$ and $\varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}$ so ${\frac {\varphi (n)}{2}}<A(n)$ fails to hold when n > 273 (actually, when n > 33). Just check a few numbers to see that n = 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21.

21 appears in the Padovan sequence, preceded by the terms 9, 12, 16 (it is the sum of the first two of these).

In science

• The atomic number of scandium.

Age 21

• In several countries 21 is the age of majority. See also: Coming of age.
• In all US states, 21 is the drinking age.
  • However, in Puerto Rico and U.S. Virgin Islands, the drinking age is 18.
• In California, Hawaii, New York, and New Jersey, 21 is the minimum age that one person may purchase cigarettes and other tobacco products.
• In some countries it is the voting age.
• In the United States, 21 is the age at which one can purchase multiple tickets to an R-rated film without providing Identifications. It is also the age to accompany one under the age of 17 as their parent or adult guardian for an R-rated movie.
• In most US states, 21 is the minimum age at which a person may gamble or enter casinos.
• In 2011, Adele named her second studio album 21, because of her age at the time.

In sports

• Twenty-one is a variation of street basketball, in which each player, of which there can be any number, plays for himself only (i.e. not part of a team); the name comes from the requisite number of baskets.
• In badminton, and table tennis (before 2001), 21 points are required to win a game.

In other fields

See also: List of highways numbered 21

Building called "21" in Zlín, Czech Republic.

Detail of the building entrance

21 is:

• The Twenty-first Amendment repealed the Eighteenth Amendment, thereby ending Prohibition.
• The number of spots on a standard cubical (six-sided) die (1+2+3+4+5+6)
• The number of firings in a 21-gun salute honoring Royalty or leaders of countries
• 21 Guns, a 2009 song by the punk-rock band Green Day
• There are 21 trump cards of the tarot deck if one does not consider The Fool to be a proper trump card.
• The standard TCP/IP port number for FTP connection
• The Twenty-One Demands were a set of demands which were sent to the Chinese government by the Japanese government of Okuma Shigenobu in 1915
• 21 Demands of MKS led to the foundation of Solidarity in Poland.
• In Israel, the number is associated with the profile 21 (the military profile designation granting an exemption from the military service)
• 21 grams is the weight of the soul, according to research by Duncan MacDougall, generally regarded as meaningless.
• The number of the French department Côte-d'Or
• The key value and highest-winning point total of the popular casino game Blackjack
• The number of shillings in a guinea.
• The number of solar rays in the flag of Kurdistan.