# 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 225

- 225 is the smallest square that starts with 2 identical digits
- 225 is the smallest octagonal square (except trivial 1)
- 225 is the smallest square whose external digits form a square and whose internal digits form a prime
- 225 is the only three-digit square with all prime digits
- 225 is the smallest square that remain a square if prefixed with 1
- 225 is the smallest square which is pandigital is some base: 225 is 3201 is base 4
- 225 is the smallest odd power that is not prime power
- 225 is the smallest sum of a set of odd numbers that contain every digit once

## Rare Properties of 225

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

## Common Properties of 225

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

The number n is *deficient* if the sum of all its positive divisors except itself is less than n.

Compare with perfect and abundant numbers.

The number n is *evil* if it has an even number of 1's in its binary expansion.

Guess what odious numbers are.

A number is *odd* if it is not divisible by 2.

Numbers that are not odd are even. Compare with another pair -- evil and odious numbers.

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}.