# 36 (number)

 ← 35 36 37 →
Cardinalthirty-six
Ordinal36th
(thirty-sixth)
Factorization22 × 32
Divisors1, 2, 3, 4, 6, 9, 12, 18, 36
Greek numeralΛϚ´
Roman numeralXXXVI
Binary1001002
Ternary11003
Senary1006
Octal448
Duodecimal3012

36 (thirty-six) is the natural number following 35 and preceding 37.

## In mathematics

36 is both the square of six, and the eighth triangular number[1] or the sum of the first eight non-zero positive integers, which makes 36 the first non-trivial square triangular number.[2] Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than 1) whose square root is also a triangular number. 36 is also the eighth refactorable number, as it has exactly nine positive divisors, and 9 is one of them;[3] in fact, it is the smallest positive integer with exactly nine divisors, which leads 36 to be the 7th highly composite number.[4] It is the sum of the fourth pair of twin-primes (17 + 19),[5] and the 18th Harshad number in decimal, as it is divisible by the sum of its digits (9).[6]

It is the smallest number ${\displaystyle n}$ with exactly eight solutions (37, 57, 63, 74, 76, 108, 114, 126) to the Euler totient function ${\displaystyle \phi (x)=n}$. Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36; hence, it is also the eighth semiperfect number.[7]

This number is the sum of the cubes of the first three positive integers and also the product of the squares of the first three positive integers.

36 is the number of degrees in the interior angle of each tip of a regular pentagram.

The number of possible outcomes (not summed) in the roll of two distinct dice.

36 is the largest numeric base that some computer systems support because it exhausts the numerals, 0–9, and the letters, A-Z. See Base 36.

The truncated cube and the truncated octahedron are Archimedean solids with 36 edges.[9]

The number of domino tilings of a 4×4 checkerboard is 36.[10]

Since it is possible to find sequences of 36 consecutive integers such that each inner member shares a factor with either the first or the last member, 36 is an Erdős–Woods number.[11]

The sum of the integers from 1 to 36 is 666 (see number of the beast).

36 is also a Tridecagonal number.[12]

## Measurements

• The number of inches in a yard (3 feet).[13]
• In the UK, a standard beer barrel is 36 UK gallons, about 163.7 litres.[13]
• three dozen, or a quarter of a gross.

## In religion

• Jewish tradition holds that the number 36 has had special significance since the beginning of time: According to the Midrash, the light created by God on the first day of creation shone for exactly 36 hours; it was replaced by the light of the Sun that was created on the Fourth Day.[16] The Torah commands 36 times to love, respect and protect the stranger.[16] Furthermore, in every generation there are 36 righteous people (the "Lamed Vav Tzadikim") in whose merit the world continues to exist.[16] In the modern celebration of Hanukkah, 36 candles are kindled in the menorah over the eight days of that holiday (not including the shamash candle).[16]
• In one Māori legend, concerning the creation of mankind by the god Tāne, 36 gods took active part in assembling the various parts of the first human before Tāne breathed life into her.[17]
• In Shaivism (s.a. Kaśmir Śaivism), The 36 tattvas describe the Absolute, its internal aspects and the creation including living beings, down to the physical reality.
• In Egyptian religion, the 36 decans are a series of gods presiding over the degrees of the zodiac and the fixed stars.[18]
• Traditional Hindu marriage bases compatibility on a set of 36 astrological qualities or Gunas, 18 of which must be shared if the union is to be considered auspicious.

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
2. ^ "Sloane's A001110 : Square triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
3. ^ Sloane, N. J. A. (ed.). "Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
4. ^ "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
5. ^ Sloane, N. J. A. (ed.). "Sequence A001097 (Twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-15.
6. ^ 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 2023-06-15.
7. ^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
8. ^ Weisstein, Eric W. "36 Officer Problem". mathworld.wolfram.com. Retrieved 2020-08-21.
9. ^ Weisstein, Eric W. "Archimedean Solid". mathworld.wolfram.com. Retrieved 2020-08-21.
10. ^ Weisstein, Eric W. "Domino Tiling". mathworld.wolfram.com. Retrieved 2020-08-21.
11. ^ "Sloane's A059756 : Erdős-Woods numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
12. ^ Sloane, N. J. A. (ed.). "Sequence A051865 (13-gonal (or tridecagonal) numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
13. ^ a b "How Many? A Dictionary of Units of Measurement. -Y". Archived from the original on 22 September 2007. Retrieved 2007-09-15.
14. ^ "WebElements.com – Krypton". Archived from the original on 4 August 2007. Retrieved 2007-08-13.
15. ^ "36bit.org". Archived from the original on 11 September 2007. Retrieved 2007-09-15.
16. ^ a b c d Winston, Pinchas (1995). The Wonderful World of Thirty-six. Mercava Productions. ISBN 0-9698032-4-9.
17. ^ "The Creation of Man". The Coming of the Maori. Retrieved 2016-04-24.
18. ^ Betz, Hans Dieter (1996). The Greek Magical Papyri in Translation. University of Chicago Press. ISBN 9780226044477.
19. ^ "Adelaide 36ers Homepage". Archived from the original on 11 August 2007. Retrieved 13 August 2007.