# 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 60

- The smallest non-abelian simple group (the alternating group on 5 elements) has order 60, in particular 60 is the smallest composite which is the order of a simple group
- 60 is the smallest product of the sides of a Pythagorean triangle
- It is possible to arrange 60 unit squares into a rectangle, so that the border squares take half the area. The only other option is with 48 squares
- 60 is the smallest number divisible by the numbers 1 to 6
- 60 is the smallest number with exactly 12 divisors

## Common Properties of 60

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.