# 120 (number)

 ← 119 120 121 →
Cardinal one hundred twenty
Ordinal 120th
(one hundred and twentieth)
Factorization 23× 3 × 5
Divisors 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Roman numeral CXX
Binary 11110002
Ternary 111103
Quaternary 13204
Quinary 4405
Senary 3206
Octal 1708
Duodecimal A012
Hexadecimal 7816
Vigesimal 6020
Base 36 3C36 The 120-cell (or hecatonicosachoron) is a convex regular 4-polytope consisting of 120 dodecahedral cells

120, read as one hundred twenty, is the natural number following 119 and preceding 121.

In English and other Germanic languages, it was also formerly known as "one hundred". This "hundred" of six score is now obsolete, but is described as the long hundred or great hundred in historical contexts.

## In mathematics

120 is the factorial of 5, and the sum of a twin prime pair (59 + 61). 120 is the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of 2 (8 + 16 + 32 + 64), and four consecutive powers of 3 (3 + 9 + 27 + 81). It is highly composite, superabundant, and colossally abundant number, with its 16 divisors being more than any number lower than it has, and it is also the smallest number to have exactly that many divisors. It is also a sparsely totient number. 120 is the smallest number to appear six times in Pascal's triangle. 120 is also the smallest multiple of 6 with no adjacent prime number, being adjacent to 119 = 7 × 17 and 121 = 112.

It is the eighth hexagonal number and the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is divisible by the first 5 triangular numbers and the first 4 tetrahedral numbers.

120 is the first multiply perfect number of order three (a 3-perfect or triperfect number). The sum of its factors (including one and itself) sum to 360; exactly three times 120. Note that perfect numbers are order two (2-perfect) by the same definition.

120 is divisible by the number of primes below it, 30 in this case. However, there is no integer which has 120 as the sum of its proper divisors, making 120 an untouchable number.

The sum of Euler's totient function φ(x) over the first nineteen integers is 120.

120 figures in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that multiplied with any of the other numbers in the sequence yields a number that is one less than a square. Leonhard Euler also searched for this number, but failed to find it, but did find a fractional number that meets the other conditions, 77748028792.

The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120 degrees.

120 is a Harshad number in base 10.

## In science

120 is the atomic number of Unbinilium, an element yet to be discovered.

## In religion

• The cubits of the height of the Temple building (II Chronicles 3:4)
• The age at which Moses died (Deut. 34:7).
• By extension, in Jewish tradition, to wish someone a long life, one says, "Live until 120"
• The number of Men of the Great Assembly who canonized the Books of the Tanakh and formulated the Jewish prayers
• The number of talents of gold that the Queen of Sheba gave to Solomon in tribute (I Kings 10:10)
• The number of princes King Darius set over his kingdom (Daniel 6:2)
• The summed weight in shekels of the gold spoons offered by each tribal prince of Israel (Num. 7:86).
• In astrology, when two planets in a person's chart are 120 degrees apart from each other, this is called a trine. This is supposed to bring good luck in the person's life.

120 is also:

## References

1. "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
2. "Sloane's A004394 : Superabundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
3. "Sloane's A004490 : Colossally abundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
4. "Sloane's A036913 : Sparsely totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
5. "Sloane's A005820 : 3-perfect numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
6. "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
7. "Astrology And The Black Man". Afro American. January 31, 1970. Retrieved December 30, 2010.
8. The Game Court, National Basketball Association, retrieved 2014-04-07.
9. Porter, Darwin; Danforth Prince (2009). Frommer's Austria. Hoboken, New Jersey: Frommer's. p. 482. ISBN 978-0-470-39897-5.
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