# 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 30

- 30 is the first sphenic number, that is the smallest number with 3 distinct prime divisors
- 30 is the largest number with the property that all smaller numbers coprime to it are prime
- Both a dodecahedron and an icosahedron have 30 edges

## Rare Properties of 30

The p-*primorial* is the product of all primes less than or equal to p. It is sometimes denoted by p#.

Compare to compositorials and factorials.

The number is called *pronic* if it is the product of two consecutive numbers.

They are twice triangular numbers.

## Common Properties of 30

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.

A number is said to be *square-free* if its prime decomposition contains no repeated factors.