# 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 1089

- The following steps always yield 1089: 1) Take any three-digit integer in which the first digit is at least two more than the last, 2) Reverse the digits, and subtract the result from the original number, 3) Add to this result the number produced by reversing its digits. For example: 732 - 237 = 495, 495 + 594 = 1089
- 1089 is the smallest number whose reverse is a non-trivial integer multiple of itself (9801 = 9*1089)

## Rare Properties of 1089

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

## Common Properties of 1089

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}.