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 12
- 12 is the smallest abundant number
- 12 is the kissing number in three dimensions
- There are 12 pentominos, the polyominoes made from 5 squares
- 12 is the only number such that it is equal to the sum of 3 raised to its digits: 12 = 31 + 32
- 12 is the smallest number such that it is equal to the sum of its digits plus the cubes of its digits: 12 = 1 + 2 + 13 + 23
- Cube has 12 edges, dodecahedron has 12 faces, icosahedron has 12 vertices
- 12 is the weight of the modular discriminant as a modular form (submitted by Qiaochu Yuan)
Rare Properties of 12
The number is called pronic if it is the product of two consecutive numbers.
They are twice triangular numbers.
Common Properties of 12
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.