# 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, 107, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the first three odd primes ([math]\displaystyle{ 3\times5\times7 }[/math]) and less than the square of the next prime (11) by > 8, for [math]\displaystyle{ n=105 }[/math], n ± 2, ± 4, and ± 8 must be prime, and n ± 6, ± 10, ± 12, and ± 14 must be composite (prime gap).^{[clarification needed]}

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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