# 105 (number)

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
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, 107, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the first three odd primes ($\displaystyle{ 3\times5\times7 }$) 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 $\displaystyle{ 0 \lt k \lt log_2(n) }$. (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 other fields

105 is also:

• A Shimano Road groupset since 1984