105 (number)

From HandWiki
Short description: Natural number
← 104 105 106 →
Cardinalone hundred five
Ordinal105th
(one hundred fifth)
Factorization3 × 5 × 7
Divisors1, 3, 5, 7, 15, 21, 35, 105
Greek numeralΡΕ´
Roman numeralCV
Binary11010012
Ternary102203
Quaternary12214
Quinary4105
Senary2536
Octal1518
Duodecimal8912
Hexadecimal6916
Vigesimal5520
Base 362X36

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, Φ105, 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

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