105 (number)

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

← 104 105 106 →
← 100 101 102 103 104 105 106 107 108 109 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
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	11010012
Ternary	102203
Senary	2536
Octal	1518
Duodecimal	8912
Hexadecimal	6916

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 (${\displaystyle 3\times 5\times 7}$ ) and less than the square of the next prime (11) by > 8, for ${\displaystyle n=105}$ , 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 ${\displaystyle n-2^{k}}$  is prime, for ${\displaystyle 0<k<log_{2}(n)}$ . (This even works up to ${\displaystyle k=8}$ , ignoring the negative sign.)

105 is the smallest integer such that the factorization of ${\displaystyle x^{n}-1}$  over Q includes non-zero coefficients other than ${\displaystyle \pm 1}$ . In other words, the 105th cyclotomic polynomial, Φ₁₀₅, is the first with coefficients other than ${\displaystyle \pm 1}$ .

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

1. "Sloane's A051624 : 12-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
2. "Sloane's A051015 : Zeisel numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
3. "Sloane's A006882 : Double factorials". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
4. Sloane, N. J. A. (ed.). "Sequence A006958 (Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.