In Mathematics
120 is the factorial of 5, and the sum of a twin prime pair (59 + 61). 120 is the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of 2 (8+16+32+64), and four consecutive powers of 3 (3 + 9 + 27 + 81). It is highly composite, superabundant, and colossally abundant number, with its 16 divisors being more than any number lower than it has, and it is also the smallest number to have exactly that many divisors. It is also a sparsely totient number. 120 is the smallest number to appear six times in Pascal's triangle. 120 is also the smallest multiple of 6 with no adjacent prime number.
It is the eighth hexagonal number and the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is divisible by the first 5 triangular numbers and the first 4 tetrahedral numbers.
120 is the first multiply perfect number of order three (a 3-perfect number, triperfect). 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 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 φ(x) over the first nineteen integers is 120.
120 figures in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that multiplied with any of the other numbers in the sequence yields a number that is one less than a square. Leonhard Euler also searched for this number, but failed to find it, but did find a fractional number that meets the other conditions, 777480 / 28792.
The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120 degrees.
120 is a Harshad number in base 10.
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