# 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 720

- 720 is the largest factorial to contain all different digits
- 720 is the smallest number with 30 divisors
- 720 is the smallest number n such that τ(n)φ(n) > prime(n)
- 720 is the smallest number n such that σ(n)
^{2} is greater than 3σ(n^{2})
- 720 is the smallest number n such that prime(n) ± n and prime(n) ± 2n are all primes
- 720 is the smallest number n where either n or n+1 is divisible by the numbers from 1 to 10
- 720 is the product of the sides and the perimeter of the smallest Pythagorean triangle
- 720 is the smallest positive number differing from the next 3 greater squares by squares (720 = 27
^{2} - 3^{2} = 28^{2} - 8^{2} = 29^{2} - 11^{2})

## Rare Properties of 720

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 720

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.

The number n is called an *apocalyptic power* if 2^{n} contains the consecutive digits 666 (in decimal).

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.

The next *Ulam* number is uniquely the sum of two earlier distinct Ulam numbers.