# 36 (number)

 ← 35 36 37 →
Cardinal thirty-six
Ordinal 36th
(thirty-sixth)
Factorization 22× 32
Divisors 1, 2, 3, 4, 6, 9, 12, 18, 36
Roman numeral XXXVI
Binary 1001002
Ternary 11003
Quaternary 2104
Quinary 1215
Senary 1006
Octal 448
Duodecimal 3012
Vigesimal 1G20
Base 36 1036

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

## In mathematics

36 is both the square of 6 and a triangular number, making it a square triangular number.[1] It is the smallest square triangular number other than 1, and it is also the only triangular number other than 1 whose square root is also a triangular number. It is also a circular number- a square number that ends with the same integer by itself (6×6=36).

It is also a 13-gonal number.[2]

It is the smallest number n with exactly 8 solutions to the equation φ(x) = n. Being the smallest number with exactly 9 divisors, 36 is a highly composite number.[3] Adding up some subsets of its divisors (e.g., 6, 12 and 18) gives 36, hence 36 is a semiperfect number.[4]

This number is the sum of a twin prime (17 + 19), 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.

36 is a Harshad number in bases 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16 (and 9 other bases).

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

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.[5]

Because 362 + 1 = 1297, a prime, which is larger than 2 × 36, 36 is a Størmer number.[6]

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

36 multiplied by any other number always produces an answer whose individual digits, in decimal, add up to a multiple of nine.

## Measurements

• The number of inches in a yard (3 feet).[7]
• In the UK, a standard beer barrel is 36 UK gallons, about 163.7 litres.[7]

## 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.[10] The Torah commands 36 times to love, respect and protect the stranger.[10] Furthermore, in every generation there are 36 righteous people (the "Lamed Vav Tzadikim") in whose merit the world continues to exist.[10] In the modern celebration of Hanukkah, 36 candles are kindled in the menorah over the 8 days of that holiday (not including the shamash candle).[10]
• 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.[11]
• 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.

## References

1. "Sloane's A001110 : Square triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
2. "Sloane's A051865 : 13-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
3. "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
4. "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
5. "Sloane's A059756 : Erdős-Woods numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
6. "Sloane's A005528 : Størmer numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
7. "How Many? A Dictionary of Units of Measurement. -Y". Archived from the original on 22 September 2007. Retrieved 2007-09-15.
8. "WebElements.com – Krypton". Archived from the original on 4 August 2007. Retrieved 2007-08-13.
9. "36bit.org". Archived from the original on 11 September 2007. Retrieved 2007-09-15.
10. Pinchas Winston (1995). The Wonderful World of Thirty-six. Mercava Productions. ISBN 0-9698032-4-9.
11. "The Creation of Man". The Coming of the Maori. Retrieved 2016-04-24.
12. "Adelaide 36ers Homepage". Archived from the original on 11 August 2007. Retrieved 2007-08-13.
13. "Robin Roberts player Profile Chronology". Baseballlibrary.com. Retrieved 2007-08-13.
14. "Giants Retire Perry's Number 36". MLB.com's Giants homepage. Retrieved 2007-08-13.