# 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 24

- 24 is the only number that is the product of all the numbers less than its square root
- 24 divides the difference between any two prime squares greater than three
- There are 24 rotations of a cube
- The diagonals of a regular hexagon divide it into 24 regions
- 24 is the largest number n such that n! has n digits
- Subtracting one from each of its divisors (except 1 and 2, but including itself) yields a prime number - 24 is the largest number with this property
- 24 is the smallest abundant factorial
- 24 is the only n greater than 1 such that the sum of squares of integers from 1 to n inclusive is itself a square. In other words, 24 is the only non-trivial solution to the cannonball problem
- Three are 24 types of jigsaw pieces including the trivial square

## Rare Properties of 24

The n-th *compositorial* is the product of the first n composite numbers.

Compositorial numbers are factorials divided by primorials.

The n-th *factorial* is the product of the first n natural numbers.

The factorial deserved an exclamation mark for its notation: k! = 1*2*3*...*k.

## Common Properties of 24

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.

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