180 (number)

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← 179 180 181 →
Cardinal one hundred eighty
Ordinal 180th
(one hundred eightieth)
Factorization 22 × 32 × 5
Divisors 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
Greek numeral ΡΠ´
Roman numeral CLXXX, clxxx
Binary 101101002
Ternary 202003
Senary 5006
Octal 2648
Duodecimal 13012
Hexadecimal B416

180 (one hundred [and] eighty) is the natural number following 179 and preceding 181.

Contents

In mathematics

180 is an abundant number, with its proper divisors summing up to 366. [1] [2] 180 is also the 11th highly composite number, [3] a positive integer with 18 divisors, more than any smaller positive integer. [4] One of the consequences of 180 having so many divisors is that it is a practical number, meaning that any positive number smaller than 180 that is not a divisor of 180 can be expressed as the sum of some of 180's divisors. 180 is a Harshad number and a refactorable number. [5]

180 is the sum of two square numbers: 122 + 62. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. 180 is an Ulam number, which can be expressed as a sum of earlier terms in the Ulam sequence only as 177 + 3. [6]

180 is a 61-gonal number, [2] while 61 is the 18th prime number.

Half a circle has 180 degrees, [7] and thus a U-turn is also referred to as a 180.

Summing Euler's totient function φ(x) over the first + 24 integers gives 180.

In binary it is a digitally balanced number, since its binary representation has the same number of zeros as ones (10110100).

A triangle has three interior angles that collectively total 180 degrees. In general, the interior angles of an -sided polygon add to degrees.

References

  1. ↑ "Positive Integers: 180".
  2. 1 2 "The Number 180". VirtueScience.com. Archived from the original on 2021-01-23. Retrieved 2009-05-29.
  3. ↑ "A002182 - OEIS". oeis.org. Retrieved 2024-11-28.
  4. ↑ Sloane, N. J. A. (ed.). "SequenceA002182(Highly composite numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30.
  5. ↑ "Refactorable numbers". On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. 2016-04-18. Retrieved 2016-04-18.
  6. ↑ "Ulam numbers". On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. 2016-04-18. Retrieved 2016-04-18.
  7. ↑ Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers . London: Penguin Group. p. 142. ISBN   0-14-026149-4.
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