# 21 (number)

Short description: Natural number
 ← 20 21 22 →
Cardinaltwenty-one
Ordinal21st
(twenty-first)
Factorization3 × 7
Divisors1, 3, 7, 21
Greek numeralΚΑ´
Roman numeralXXI
Binary101012
Ternary2103
Quaternary1114
Quinary415
Senary336
Octal258
Duodecimal1912
Vigesimal1120
Base 36L36

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

The current century is the 21st century AD, under the Gregorian calendar.

## In mathematics

21 is:

• a composite number, its proper divisors being 1, 3 and 7, and a deficient number as the sum of these divisors is less than the number itself.
• a Fibonacci number as it is the sum of the preceding terms in the sequence, 8 and 13.
• the fifth Motzkin number.
• a triangular number, because it is the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21).
• an octagonal number.
• a Padovan number, preceded by the terms 9, 12, 16 (it is the sum of the first two of these) in the padovan sequence.
• a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.
• the sum of the divisors of the first 5 positive integers (i.e., 1 + (1 + 2) + (1 + 3) + (1 + 2 + 4) + (1 + 5))
• the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number.
• a Harshad 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 $\displaystyle{ \tfrac{a}{b} }$ and $\displaystyle{ \tfrac{b}{a} }$ 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 $\displaystyle{ A(n) }$ denote the quantity of the numbers smaller than n that only have factor 2 and 5 and that are coprime to n, we instantly have $\displaystyle{ \frac{\varphi(n)}{2} \lt A(n) }$.

We can easily see that for sufficiently large n, $\displaystyle{ A(n) \sim \frac{\log_2(n) \log_5(n)}{2} = \frac{\ln^{2}(n)}{2 \ln(2) \ln(5)} }$, but $\displaystyle{ \varphi(n) \sim \frac {n} {e^\gamma\; \ln \ln n} }$, $\displaystyle{ A(n) = o(\varphi(n)) }$ as n goes to infinity, thus $\displaystyle{ \frac{\varphi(n)}{2} \lt A(n) }$ fails to hold for sufficiently large n.

In fact, For every n > 2, we have

$\displaystyle{ A\lt 1 + \log_2(n) + \frac{3 \log_5(n)}{2} + \frac{\log_2(n) \log_5(n)}{2} }$

and

$\displaystyle{ \varphi(n) \gt \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}} }$

so $\displaystyle{ \frac{\varphi(n)}{2} \lt }$ fails to hold when n > 273 (actually, when n > 33).

Just check a few numbers to see that '= 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21.

## Age 21

• In thirteen countries, 21 is the age of majority. See also: Coming of age.
• In eight countries, 21 is the minimum age to purchase tobacco products.
• In seventeen countries, 21 is the drinking age.
• In nine countries, it is the voting age.
• In the United States:
• 21 is the minimum age at which a person may gamble or enter casinos in most states (since alcohol is usually provided).
• 21 is the minimum age to purchase a handgun or handgun ammunition under federal law.
• 21 is the age at which one can purchase multiple tickets to an R-rated film.
• In some states, 21 is the minimum age to accompany a learner driver, provided that the person supervising the learner has held a full driver license for a specified amount of time. See also: List of minimum driving ages.

## 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 three-on-three basketball games held under FIBA rules, branded as 3x3, the game ends by rule once either team has reached 21 points.
• In badminton, and table tennis (before 2001), 21 points are required to win a game.
• In AFL Women's, the top-level league of women's Australian rules football, each team is allowed a squad of 21 players (16 on the field and five interchanges).

## In other fields

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
• "Twenty One", a 1994 song by an Irish rock band The Cranberries
• "21 Guns", a 2009 song by the punk-rock band Green Day
• Twenty One Pilots, an American musical duo
• 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 China 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)
• Duncan MacDougall reported that 21 grams is the weight of the soul, according to an experiment.
• The number of the French department Côte-d'Or
• Twenty-One (card game), an ancient card game in which the key value and highest-winning point total is 21
• Blackjack, a modern version of Twenty-One played in casinos
• The number of shillings in a guinea.
• The number of solar rays in the flag of Kurdistan.
• Twenty-One, an American game show that became the center of the 1950s quiz show scandals when it was shown to be rigged.
• The number on the logo for the American game show Catch 21
• The total number of Bitcoin to be released is 21 million “
• Twenty-One, a 1991 British-American drama film directed by Don Boyd and starring Patsy Kensit.