# 105 (number)

__: Natural number__

**Short description**
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Cardinal | one hundred five | |||

Ordinal | 105th (one hundred fifth) | |||

Factorization | 3 × 5 × 7 | |||

Divisors | 1, 3, 5, 7, 15, 21, 35, 105 | |||

Greek numeral | ΡΕ´ | |||

Roman numeral | CV | |||

Binary | 1101001_{2} | |||

Ternary | 10220_{3} | |||

Quaternary | 1221_{4} | |||

Quinary | 410_{5} | |||

Senary | 253_{6} | |||

Octal | 151_{8} | |||

Duodecimal | 89_{12} | |||

Hexadecimal | 69_{16} | |||

Vigesimal | 55_{20} | |||

Base 36 | 2X_{36} |

**105** (**one hundred [and] five**) is the natural number following 104 and preceding 106.

## In mathematics

105 is a triangular number, a dodecagonal number^{[1]} and the first Zeisel number.^{[2]} It is the first odd sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7.^{[3]} It is also the sum of the first five square pyramidal numbers.

105 comes in the middle of the prime quadruplet (101, 103, 107, 109). The only other such numbers less than a thousand are 9, 15, 195 and 825.

105 is also the middle of the only prime sextuplet (97, 101, 103, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the 1st 3 odd primes (3,5,7), and less than the square of the next prime, 11, by > 8, for n=105, n ± 2, ± 4, and ± 8 must be prime and n ± 6, ± 10, ± 12, and ± 14 must be composite (prime gap).

105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83 and 97. The distinct prime factors of 105 add up to 15, and so do those of 104, hence the two numbers form a Ruth-Aaron pair under the first definition.

105 is also a number *n* for which [math]\displaystyle{ n - 2^k }[/math] is prime, for [math]\displaystyle{ 0 \lt k \lt log_2(n) }[/math]. (This even works up to [math]\displaystyle{ k = 8 }[/math], ignoring the negative sign.)

105 is the smallest integer such that the factorization of [math]\displaystyle{ x^n-1 }[/math] over **Q** includes non-zero coefficients other than [math]\displaystyle{ \pm 1 }[/math]. In other words, the 105th cyclotomic polynomial, Φ_{105}, is the first with coefficients other than [math]\displaystyle{ \pm 1 }[/math].

105 is the number of parallelogram polyominoes with 7 cells.^{[4]}

## In science

- The atomic number of dubnium.

## In other fields

105 is also:

- A Shimano Road groupset since 1984

## See also

- List of highways numbered 105

## References

- Wells, D.
*The Penguin Dictionary of Curious and Interesting Numbers*London: Penguin Group. (1987): 134

- ↑ "Sloane's A051624 : 12-gonal numbers". OEIS Foundation. https://oeis.org/A051624.
- ↑ "Sloane's A051015 : Zeisel numbers". OEIS Foundation. https://oeis.org/A051015.
- ↑ "Sloane's A006882 : Double factorials". OEIS Foundation. https://oeis.org/A006882.
- ↑ Sloane, N. J. A., ed. "Sequence A006958 (Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused))". OEIS Foundation. https://oeis.org/A006958.

Original source: https://en.wikipedia.org/wiki/105 (number).
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