# 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 625

- 625 is the smallest square that remain squares when prefixed with a 5
- 625 is the largest three-digit automorphic number

## Rare Properties of 625

The number n is called an *automorphic* number if (the decimal expansion of) n^{2} ends with n. These numbers are also called *curious*.

It is curious, how for a k-digit automorphic number n there is another automorphic number -- 10^{k} + 1 - n. For this to work with n=1, you have to treat 1 as a zero-digit number.

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

## Common Properties of 625

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.

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.

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

Guess what evil 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}.