- Twelve is a composite
number, the smallest number with exactly six divisors,
its divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one
being twenty-four. Twelve is also
a superior highly composite
number, the next one being sixty.
It is the first composite number of the form
*p*^{2}*q*; a square-prime, and also the first member of the (*p*^{2}) family in this form. 12 has an aliquot sum of 16 (133% in abundance). Accordingly, 12 is the first abundant number (in fact a superabundant number) and demonstrates an 8-member aliquot sequence; {12,16,15,9,4,3,1,0} 12 is the 3rd composite number in the 3-aliquot tree. The only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant (18 and 20). - Twelve is a sublime
number, a number that has a perfect
number of divisors, and the sum
of its divisors is also a perfect number.
^{[11]}Since there is a subset of 12's proper divisors that add up to 12 (all of them but with 4 excluded), 12 is a semiperfect number. - If an odd perfect
number is of the form 12
*k*+ 1, it has at least twelve distinct prime factors. - Twelve is a superfactorial,
being the product of the first three factorials.
^{}Twelve being the product of three and four, the first four positive integers show up in the equation 12 = 3 × 4, which can be continued with the equation 56 = 7 × 8. - Twelve is the ninth Perrin
number, preceded in the sequence by 5, 7, 10,
^{[14]}and also appears in the Padovan sequence, preceded by the terms 5, 7, 9 (it is the sum of the first two of these).^{[15]}It is the fourth Pell number, preceded in the sequence by 2 and 5 (it is the sum of the former plus twice the latter). - Twelve is probably the last even number that is the sum of only one pair of prime numbers (5+7). (Goldbach's conjecture)
- A twelve-sided polygon is a dodecagon.
A twelve-faced polyhedron is a dodecahedron.
Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. Twelve is a pentagonal number. The densest
three-dimensional lattice sphere packing has each sphere touching 12 others,
and this is almost certainly true for
*any*arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions. - Twelve
is the smallest weight for which a cusp
form exists. This cusp form is
the discriminant Δ(
*q*) whose Fourier coefficients are given by the Ramanujan τ-function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function. This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at −1 i.e. ζ(−1) = −1/12, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons. - There are twelve Jacobian elliptic functions and twelve cubic distance-transitive graphs.
- There are 12 Latin squares of size 3 × 3.
- The duodecimal system (12
_{10}[twelve] = 10_{12}), which is the use of 12 as a division factor for many ancient and medieval weights and measures, including hours, probably originates from Mesopotamia. - In base thirteen and higher bases (such as hexadecimal), twelve is represented as C. In base 10, the number 12 is a Harshad number.

*Source:*

*https://en.wikipedia.org/wiki/12_(number)*

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