# 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 1024

- 1024 is the number of bytes in a kilobyte
- 1024 is the smallest number which uses a proper subset of its digits in its prime factorization: 1024 = 2
^{10} — the factorization uses 1, 0, 2, and doesn't use 4
- 1024 is the smallest square with 4 different digits
- 1024 is the smallest power of 2 with the digit 0 in its decimal expansion
- 1024 is the smallest square that is concatenation of two legs of a Pythagorean triangle (10
^{2} + 24^{2} = 26^{2})

## Rare Properties of 1024

A number is a *power of 2* if it is 2 to some power.

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

## Common Properties of 1024

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 *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 *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}.

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