10

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10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.

Ten is the fifth composite number. It is also the smallest noncototient, a number that cannot be expressed as the difference between any integer and the total number of coprimes below it.^[1] It is the second discrete semiprime (), as well as the second member of the discrete semiprime family. Ten is the only number whose sum and difference of its prime divisors yield prime numbers ( and ). In general, powers of 10 contain divisors, where is the number of digits: 10 has 2² = 4 divisors, 100 has 3² = 9 divisors, 1,000 has 4² = 16 divisors, 10,000 has 5² = 25 divisors, and so forth. Ten is the smallest number whose status as a possible friendly number is unknown.^[2]${\displaystyle 2\times 5}$${\displaystyle 2\times q}$${\displaystyle 2+5=7}$${\displaystyle 5-2=3}$${\displaystyle n^{2}}$${\displaystyle n}$

The factorial of ten is equal to the product of the factorials of the first three odd primes, .^[4] Ten is also the first number whose fourth power can be written as a sum of two squares in two different ways ( and ).${\displaystyle 10!=3!\cdot 5!\cdot 7!}$${\displaystyle 80^{2}+60^{2}}$${\displaystyle 96^{2}+28^{2}}$

Ten has an aliquot sum σ(10) of 8 and is accordingly the first discrete semiprime to be in deficit, with all subsequent discrete semiprimes in deficit.^[5] The aliquot sequence for 10 comprises five members (10, 8, 7, 1, 0) with this number being the second composite member of the 7-aliquot tree.^[6] 10 is also a Perrin number, preceded in the sequence by (5, 5, 7).^[7]In the sequence of triangular numbers, indexed powers of 10 in this sequence generate the following sequence of triangular numbers in decimal: 55 (10th), 5,050 (100th), 500,500 (1,000th), ...^[8][a]

While 55 is the tenth triangular number, it is also the tenth Fibonacci number, and the largest such number to also be a triangular number.^[9] Ten is also the first non-trivial decagonal number,^[10] the third centered triangular number^[11] and tetrahedral number,^[12] and the fifth semi-meandric number.^[13]

10 is the fourth telephone number, and the number of Young tableaux with four cells.^[14] It is the number of -queens problem solutions for .^[15]${\displaystyle n}$${\displaystyle n=5}$

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