12

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Number

$12$ (twelve) is:

$2^2 \times 3$


The base of the duodecimal number system


The number of edges of the regular octahedron and its dual, the cube


The number of distinct pentominoes, up to reflection


The $1$st power of $12$ after the zeroth $1$:
$12 = 12^1$


The $1$st of three $2$-digit integers divisible by both the sum and product of its digits:
$12 = \left({1 + 2}\right) \times 4 = \left({1 \times 2}\right) \times 6$


The $1$st abundant number:
$1 + 2 + 3 + 4 + 6 = 16 > 12$


The $2$nd semiperfect number after $6$:
$12 = 2 + 4 + 6$


The $3$rd pentagonal number after $1$, $5$:
$12 = 1 + 4 + 7 = \dfrac {3 \left({3 \times 3 - 1}\right)} 2$


The $3$rd superfactorial after $1$, $2$:
$12 = 3\$ = 3! \times 2! \times 1!$


The $4$th special highly composite number after $1$, $2$, $6$


The $5$th highly composite number after $1$, $2$, $4$, $6$:
$\tau \left({12}\right) = 6$


The $5$th superabundant number after $1$, $2$, $4$, $6$, and the smallest which is also abundant:
$\dfrac {\sigma \left({12}\right)} {12} = \dfrac {28} {12} = 2 \cdotp \dot 3$


The $5$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
$3$, $4$, $6$, $7$, $12$


The $5$th generalized pentagonal number after $1$, $2$, $5$, $7$:
$12 = \dfrac {3 \left({3 \times 3 - 1}\right)} 2$


The $5$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $\ldots$


The $6$th even number after $2$, $4$, $6$, $8$, $10$ which cannot be expressed as the sum of $2$ composite odd numbers


The $7$th positive integer after $1$, $2$, $3$, $4$, $6$, $8$ such that all smaller positive integers coprime to it are prime


The $7$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $\ldots$


The $8$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$:
$\sigma \left({12}\right) = 28$


The $8$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes


The $9$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$, $10$ which cannot be expressed as the sum of exactly $5$ non-zero squares.


The $11$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$:
$12 = 6 \times 2 = 6 \times \left({1 \times 2}\right)$


The $11$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$:
$12 = 4 \times 3 = 4 \times \left({1 + 2}\right)$


The square of the reversal of $12$ equals the reversal of the square of $12$:
$12^2 = 144$
$21^2 = 441$


$12 = 3 \times 4$, and $56 = 7 \times 8$


Also see



Historical Note

There are many occurrences of the number $12$ in various cultures in history:

There are $12$ months in a year, divided into $4$ seasons of $3$ months each.
There are $12$ signs of the Zodiac, divided into $3$ sets of $4$ each, and categorising them another way, into $4$ sets of $3$ each.
There are $12$ hours in each day and in each night.
There are $12$ semitones in an octave of a well-tempered scale.
There are $12$ inches in a foot.
There were $12$ pence in $1$ shilling in pre-decimal British coinage.


The word dozen, now falling into disuse, means a set of $12$.


Sources