# 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 1000

- 1000 is the smallest cube formed by concatenating other cubes
- 1000 = 10
^{3}, as such it is the smallest product of three 2-digit numbers
- 1000 is the smallest number containing letter A when written out in American English
- 1000 is the smallest number whose scientific notation (1E3) is shorter than its decimal representation
- 1000 is the smallest cube of a non-palindromic number

## Rare Properties of 1000

The number n is a *cube* if it is the cube of an integer.

## Common Properties of 1000

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.

One can take the sum of the squares of the digits of a number. Those numbers are *happy* for which iterating this operation eventually leads to 1.

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.