# 6

 ← 5 6 7 →
Cardinalsix
Ordinal6th
(sixth)
Numeral systemsenary
Factorization2 × 3
Divisors1, 2, 3, 6
Greek numeralϚ´
Roman numeralVI
Roman numeral (unicode)Ⅵ, ⅵ, ↅ
Greek prefixhexa-/hex-
Latin prefixsexa-/sex-
Binary1102
Ternary203
Quaternary124
Quinary115
Senary106
Octal68
Duodecimal612
Vigesimal620
Base 36636
Greekστ (or ΣΤ or ς)
Arabic & Kurdish٦
Persian۶
Urdu
Amharic
Bengali
Chinese numeral六，陸
Devanāgarī
Hebrewו
Khmer
Thai
Telugu
Tamil
Saraiki٦

6 (six) is the natural number following 5 and preceding 7.

The SI prefix for 10006 is exa- (E), and for its reciprocal atto- (a).

## In mathematics

6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number; its proper divisors are 1, 2 and 3.

Since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and ${\displaystyle {\mathcal {S}}}$-perfect number.[1][2]

As a perfect number:

Six is the only number that is both the sum and the product of three consecutive positive numbers.[4]

Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler".[5] Six is a congruent number.[6]

Six is the first discrete biprime (2 × 3) and the first member of the (2 × q) discrete biprime family.

Six is a unitary perfect number,[7] a primary pseudoperfect number,[8] a harmonic divisor number[9] and a superior highly composite number, the last to also be a primorial. The next superior highly composite number is 12. The next primorial is 30.

There are no Graeco-Latin squares with order 6. If n is a natural number that is not 2 or 6, then there is a Graeco-Latin square with order n.

There is not a prime p such that the multiplicative order of 2 modulo p is 6, that is, ordp(2) = 6. By Zsigmondy's theorem, if n is a natural number that is not 1 or 6, then there is a prime p such that ordp(2) = n. See A112927 for such p.

The ring of integer of the sixth cyclotomic field Q6) , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where ${\displaystyle \omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}}$.

The smallest non-abelian group is the symmetric group S3 which has 3! = 6 elements.

S6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n isomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in one-to-one correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n = 6.

Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.

A cube has 6 faces

6 is the largest of the four all-Harshad numbers.

A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of a power of 2 (namely 21) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon.

Six is also an octahedral number.[10] It is a triangular number and so is its square (36).

There are six basic trigonometric functions.

There are six convex regular polytopes in four dimensions.

The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.

All primes above 3 are of the form 6n ± 1 for n ≥ 1.

### List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 50 100 1000
6 × x 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 150 300 600 6000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
6 ÷ x 6 3 2 1.5 1.2 1 0.857142 0.75 0.6 0.6 0.54 0.5 0.461538 0.428571 0.4
x ÷ 6 0.16 0.3 0.5 0.6 0.83 1 1.16 1.3 1.5 1.6 1.83 2 2.16 2.3 2.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
6x 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336 13060694016
x6 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 2985984 4826809
Other Languages
አማርኛ: ስድስት
Ænglisc: 6 (getæl)
العربية: 6 (عدد)
ܐܪܡܝܐ: 6 (ܡܢܝܢܐ)
অসমীয়া:
asturianu: Seis
Avañe'ẽ: Poteĩ
авар: Анлъго
azərbaycanca: 6 (ədəd)
تۆرکجه: ۶ (سایی)
Bân-lâm-gú: 6
беларуская: 6 (лік)
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བོད་ཡིག: ༦ (གྲངས་ཀ།)
català: Sis
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čeština: 6 (číslo)
chiShona: Tanhatu
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Cymraeg: Chwech
dansk: 6 (tal)
Deitsch: Sex
Deutsch: Sechs
eesti: Kuus
Ελληνικά: 6 (αριθμός)
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español: Seis
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føroyskt: 6 (tal)
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galego: Seis
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한국어: 6
Hausa: Shida
հայերեն: Վեց
hrvatski: Šest
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Igbo: Isii
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interlingua: 6 (numero)
Iñupiak: Itchaksrat
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коми: 6 (квайт)
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лакку: Ряхва
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latviešu: 6 (skaitlis)
Lëtzebuergesch: Sechs
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Limburgs: Zès
lingála: Motóbá
Luganda: Mukaaga
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Nāhuatl: Chicuacē
Na Vosa Vakaviti: Ono
Nederlands: 6 (getal)

Napulitano: Seie
Nordfriisk: Seeks
norsk: 6 (tall)
norsk nynorsk: Talet 6
ଓଡ଼ିଆ: ୬ (ସଂଖ୍ୟା)
oʻzbekcha/ўзбекча: 6 (son)
پنجابی: 6
پښتو: ۶
Перем Коми: 6 (квать)
polski: 6 (liczba)
português: Seis
română: 6 (cifră)
Runa Simi: Suqta
русский: 6 (число)
саха тыла: Алта (ахсаан)
ᱥᱟᱱᱛᱟᱲᱤ:
Gagana Samoa: 6 (numera)
Scots: 6 (nummer)
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ślůnski: 6 (nůmera)
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کوردی: ٦ (ژمارە)
Sranantongo: Numro 6
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Türkçe: 6 (sayı)
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Xitsonga: Tsevu
ייִדיש: 6 (נומער)