Number Gossip
(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)
Unique Properties of 120
- 120 is the smallest 3-perfect number — the sum of its divisors is equal to itself times 3
- 120 is the largest integer n for which the number of primes less than or equal to n equals n/4
- 120 is the smallest number to appear 6 times in Pascal's triangle
- The internal angles of a regular hexagon are all 120 degrees
- 120 is the number of 1-piece positions at checkers
- 120 is the smallest triangular number whose digit permutations yield at least two other triangular numbers (120 yields 21 and 210)
- 120 is the smallest number that can be represented as a product of consecutive integers in more than one way: 120 = 2*3*4*5 = 4*5*6
- 120 is the smallest even triangular number followed by a composite number
Rare Properties of 120
The n-th factorial is the product of the first n natural numbers.
The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.
A tetrahedral number is the number of balls you can put in a triangular pyramid.
This is the space generalization of triangular and square numbers.
Common Properties of 120
The number n is abundant if the sum of all its positive divisors except itself is more than n.
They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.
A positive integer greater than 1 that is not prime is called composite.
Composite numbers are opposite to prime numbers.
A number is even if it is divisible by 2.
Numbers that are not even are odd. Compare with another pair -- evil and odious numbers.
The number n is evil if it has an even number of 1's in its binary expansion.
Guess what odious numbers are.
The number n is practical if all numbers strictly less than n are sums of distinct divisors of n.
If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a triangle number.
Compare to square, pentagonal and tetrahedral numbers.
The untouchable numbers are those that are not the sum of the proper divisors of any number.