# 72 (number)

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 ← 71 72 73 →
Cardinalseventy-two
Ordinal72nd
(seventy-second)
Factorization23 × 32
Divisors1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Greek numeralΟΒ´
Roman numeralLXXII
Binary10010002
Ternary22003
Senary2006
Octal1108
Duodecimal6012

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).

## In mathematics

Seventy-two is a pronic number, as it is the product of 8 and 9.

72 is an abundant number,[1] with a total of 12 factors, and a Euler totient of 24.[2] 72 is also a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer below 72.[3] It is equal to the sum 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.[3] 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.[4] 72 is also a Harshad number in decimal, as it is divisible by the sum of its digits.[5]

• 72 is the smallest Achilles number, as it's a powerful number that is not itself a power.[6]
• 72 is the sum of four consecutive primes (13 + 17 + 19 + 23), as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19).
• 72 is the smallest number whose fifth power is the sum of five smaller fifth powers: 195 + 435 + 465 + 475 + 675 = 725.[7]

72 is the number of degrees in the central angle of a regular pentagon, which is constructible with a compass and straight-edge.

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 ${\displaystyle {\tilde {T}}_{9}}$: 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 ${\displaystyle \mathrm {E} _{8}}$ lattice. Furthermore, ${\displaystyle {\tilde {T}}_{9}}$ shares the same fundamental symmetries with the Coxeter-Dynkin over-extended form ${\displaystyle \mathrm {E} _{8}}$++ equivalent to the tenth-dimensional symmetries of Lie algebra ${\displaystyle \mathrm {E} _{10}}$.

72 lies between the eighth 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, with all primes greater than or equal to 73 non-supersingular. Sporadic groups are a family of twenty-six finite simple groups, with ${\displaystyle \mathrm {E} _{6}}$, ${\displaystyle \mathrm {E} _{7}}$, and ${\displaystyle \mathrm {E} _{8}}$ representing 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.

## In religion

• The number of languages spoken at the Tower of Babylon, according to later tradition.
• The conventional number of scholars translating the Septuagint, according to the legendary account in the "Letter of Aristeas".
• The conventional number of disciples sent forth by Jesus in Luke 10 in some manuscripts (seventy in others).
• The number of names of God, according to Kabbalah (see names of God in Judaism).
• The Shemhamphorasch related to the number of the names of God.
• The total number of books in the Bible in the Catholic version if the Book of Lamentations is considered part of the Book of Jeremiah.
• The current distribution of the Book of Revelation is 22 chapters, adopted since the 13th century, but the oldest known division of the text is that of the Greek commentator Andrew of Cesary (6th century) in 72 chapters.
• The number of people martyred along with Imam Hussain at the Battle of Karbala.
• The number of houri each Muslim martyr (or every Muslim male, according to some ahadith) shall receive as companions in Paradise.
• The degrees of the Jacob's Ladder were to the number of 72, according to the Zohar.
• The 72 disciples of Confucius who mastered his teachings (also given as 77).
• Mahavira, the twenty-fourth and last tirthankara of Jainism, is said to have attained nirvana after his physical death at the age of 72.
• Thoth, in an Egyptian creation myth, wins a 72nd of each day of the year from the Moon in a game of draughts, as a favour to Nut, the Sky Goddess. He uses these portions to make the five intercalary days on which the remaining Gods and Goddesses are born.[9]
• The good god Osiris was enclosed in a coffin by 72 evil disciples and accomplices of Set.[10]
• At the age of the puberty, the young Parsee received the investiture of the sacred cord Kucti made of 72 linens in symbol of the community.
• In Cao Đài, the number of planets between hell and heaven.
• There are 72 stupas which comprise Borobudur, the world's largest Buddhist temple.
• 72 major temples have been found at Angkor, seat of the ancient Khmer Empire.
• In Islam, 72 is the number of sects or denominations that are doomed to Hell, according to Hadith (Sayings of prophet Muhammad).[11][12]
• The number of demons sealed away by King Solomon with The Lesser Key of Solomon.

## In sports and games

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Source: "72 (number)", Wikipedia, Wikimedia Foundation, (2022, November 25th), https://en.wikipedia.org/wiki/72_(number).

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###### Footnotes
1. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
2. ^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
3. ^ a b Sloane, N. J. A. (ed.). "Sequence A097942 (Highly totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
4. ^ Sloane, N. J. A. (ed.). "Sequence A002088 (Sum of totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
5. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
6. ^ Sloane, N. J. A. (ed.). "Sequence A052486 (Achilles numbers - powerful but imperfect.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
7. ^ David Wells: The Penguin Dictionary of Curious and Interesting Numbers
8. ^ Sloane, N. J. A. (ed.). "Sequence A005418 (...row sums of Losanitsch's triangle.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-22.
9. ^
10. ^ "Egyptian Myths", George Hart, p41, University of Texas Press, 1990
11. ^ "Sects In Islam - 73 Groups in Islam, Division - Denominations". Archived from the original on 6 May 2013. Retrieved 24 March 2013.
12. ^ Sunan Ibn Maajah, no. 3982 "My Ummah will be divided into seventy-three sects, one of which will be in Paradise and seventy-two will be in the Fire"
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