# 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 108

- 108 is 3 hyperfactorial, that is the product of the first three numbers raised to themselves: 108 = 1
^{1} * 2^{2} * 3^{3}
- 108 is the smallest number whose divisors contain every digit at least once
- 108 is the smallest n such that n-th Fibonacci number is divisible by n, n + 1 and n - 1
- The interior angles of an equilateral pentagon measure 108 degrees each
- A web page on 108: Why 108? Article on the symbolism of the number 108

## Common Properties of 108

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.

An integer n is *powerful* if for every prime p dividing n, p^{2} also divides n.

How much power? They all can be written as a^{2} b^{3}.

The number n is *practical* if all numbers strictly less than n are sums of distinct divisors of n.