| ||||
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Cardinal | seventy-two | |||
Ordinal | 72nd (seventy-second) | |||
Factorization | 23 × 32 | |||
Divisors | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | |||
Greek numeral | ΟΒ´ | |||
Roman numeral | LXXII | |||
Binary | 10010002 | |||
Ternary | 22003 | |||
Senary | 2006 | |||
Octal | 1108 | |||
Duodecimal | 6012 | |||
Hexadecimal | 4816 |
72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or 6 dozen (i.e., 60 in duodecimal).
Seventy-two is a pronic number, as it is the product of 8 and 9. [1] It is the smallest Achilles number, as it's a powerful number that is not itself a power. [2]
72 is an abundant number. [3] With exactly twelve positive divisors, including 12 (one of only two sublime numbers), [4] 72 is also the twelfth member in the sequence of refactorable numbers. [5] 72 has a Euler totient of 24, [6] which makes it a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72. [7] It is equal to the sum of its preceding smaller highly totient numbers 24 and 48, and contains the first six highly totient numbers 1, 2, 4, 8, 12 and 24 as a subset of its proper divisors. 144, or twice 72, is also highly totient, as is 576, the square of 24. [7] While 17 different integers have a totient value of 72, the sum of Euler's totient function φ(x) over the first 15 integers is 72. [8] It also is a perfect indexed Harshad number in decimal (twenty-eighth), as it is divisible by the sum of its digits (9). [9]
72 plays a role in the Rule of 72 in economics when approximating annual compounding of interest rates of a round 6% to 10%, due in part to its high number of divisors.
Inside Lie algebras:
There are 72 compact and paracompact Coxeter groups of ranks four through ten: 14 of these are compact finite representations in only three-dimensional and four-dimensional spaces, with the remaining 58 paracompact or noncompact infinite representations in dimensions three through nine. These terminate with three paracompact groups in the ninth dimension, of which the most important is : it contains the final semiregular hyperbolic honeycomb 621 made of only regular facets and the 521 Euclidean honeycomb as its vertex figure, which is the geometric representation of the lattice. Furthermore, shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form ++ equivalent to the tenth-dimensional symmetries of Lie algebra .
72 lies between the 8th pair of twin primes (71, 73), where 71 is the largest supersingular prime that is a factor of the largest sporadic group (the friendly giant ), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers [21] [lower-alpha 1] that is also the number of distinct orders (without multiplicity) inside all 194 conjugacy classes of . [22] Sporadic groups are a family of twenty-six finite simple groups, where , , and are associated exceptional groups that are part of sixteen finite Lie groups that are also simple, or non-trivial groups whose only normal subgroups are the trivial group and the groups themselves. [lower-alpha 2]
Seventy-two is also:
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.
90 (ninety) is the natural number following 89 and preceding 91.
22 (twenty-two) is the natural number following 21 and preceding 23.
24 (twenty-four) is the natural number following 23 and preceding 25.
27 is the natural number following 26 and preceding 28.
57 (fifty-seven) is the natural number following 56 and preceding 58.
58 (fifty-eight) is the natural number following 57 and preceding 59.
63 (sixty-three) is the natural number following 62 and preceding 64.
104 is the natural number following 103 and preceding 105.
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
360 is the natural number following 359 and preceding 361.
144 is the natural number following 143 and preceding 145.
700 is the natural number following 699 and preceding 701.
135 is the natural number following 134 and preceding 136.
100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.
177 is the natural number following 176 and preceding 178.
1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.
744 is the natural number following 743 and preceding 745.
888 is the natural number following 887 and preceding 889.