12

== 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:
::$12 = \paren {1 + 2} \times 4 = \paren {1 \times 2} \times 6$
: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 $1$, $5$:
::$12 = 1 + 4 + 7 = \dfrac {3 \paren {3 \times 3 - 1} } 2$
:The $3$rd superfactorial after $1$, $2$:
::$12 = 3\$ = 3! \times 2! \times 1!$
:The $3$rd positive integer after $6$, $9$ whose cube can be expressed as the sum of $3$ positive cube numbers:
::$12^3 = 6^3 + 8^3 + 10^3$
:The $4$th special highly composite number after $1$, $2$, $6$
:The $4$th of $6$ integers after $2$, $5$, $8$ which cannot be expressed as the sum of distinct triangular numbers
:The $5$th highly composite number after $1$, $2$, $4$, $6$:
::$\map {\sigma_0} {12} = 6$
:The $5$th superabundant number after $1$, $2$, $4$, $6$, and the smallest which is also abundant:
::$\dfrac {\map {\sigma_1} {12} } {12} = \dfrac {28} {12} = 2 \cdotp \dot 3$
: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 $1$, $2$, $5$, $7$:
::$12 = \dfrac {3 \paren {3 \times 3 - 1} } 2$
:The $5$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $\ldots$
:The number of distinct free pentominoes
: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 $1$, $2$, $3$, $4$, $6$, $8$, $10$:
::$\map {\sigma_1} {12} = 28$
: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 $10$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
:The $10$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The $11$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$:
::$12 = 6 \times 2 = 6 \times \paren {1 \times 2}$
:The $11$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$:
::$12 = 4 \times 3 = 4 \times \paren {1 + 2}$
: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 ==
* 12 Pentominoes
* Product of Proper Divisors of 12
* 12 times Divisor Sum of 12 equals 14 times Divisor Sum of 14
* 8 Mutually Non-Attacking Queens on Chessboard
* 12 Knights to Attack or Occupy All Squares on Chessboard
* Twelve Factorial plus One is divisible by 13 Squared
* Square of Reversal of Small-Digit Number
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== Historical Note ==
== Sources ==
* : $12$
* : $12$
Category:Specific Numbers
Category:12