# 21 (number)

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Cardinaltwenty-one
Ordinal21st
(twenty-first)
Factorization3 × 7
Divisors1, 3, 7, 21
Greek numeralΚΑ´
Roman numeralXXI
Binary101012
Ternary2103
Quaternary1114
Quinary415
Senary336
Octal258
Duodecimal1912
Hexadecimal1516
Vigesimal1120
Base 36L36

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, because it is the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21).
• an octagonal number.
• a composite number, its proper divisors being 1, 3 and 7.
• 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}} .

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}} 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}} 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).

## 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 identification. 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 the United Kingdom, 21 is the minimum age to accompany a learner driver, provided that the person supervising the learner has held a full driving licence for a minimum of three years.

21 is: