120 (number)

120 (one hundred [and] twenty) is the natural number following 119 and preceding 121. In the Germanic languages, the number 120 was also formerly known as "one hundred". This "hundred" of six score is now obsolete, but is described as the long hundred or great hundred in historical contexts. 120 is the factorial of 5, i.e., . the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is the smallest number to appear six times in Pascal's triangle (as all triangular and tetragonal numbers appear in it). Because 15 is also triangular, 120 is a doubly triangular number. 120 is divisible by the first 5 triangular numbers and the first 4 tetrahedral numbers. It is the eighth hexagonal number. highly composite, superior highly composite, superabundant, and colossally abundant number, with its 16 divisors, it is the smallest number to have that many divisors. It is also a sparsely totient number. 120 is also the smallest highly composite number as well as the first multiple of six with no adjacent prime number, being adjacent to and . 120 is the first multiply perfect number of order three (a 3-perfect or triperfect number). The sum of its factors (including one and itself) sum to 360; exactly three times 120. Note that perfect numbers are order two (2-perfect) by the same definition. 120 is the sum of a twin prime pair (59 + 61) and the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of two (8 + 16 + 32 + 64), and four consecutive powers of three (3 + 9 + 27 + 81). 120 is divisible by the number of primes below it, 30 in this case. However, there is no integer which has 120 as the sum of its proper divisors, making 120 an untouchable number. The sum of Euler's totient function over the first nineteen integers is 120. As 120 is a factorial and one less than a square (), it, with 11, is one of the few Brown number pairs. 120 appears in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120.