# 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 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

## 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.