# 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 10

- 10 is the base of our number system
- In base n, n is always written "10"
- 10 is the only semi-prime number with the property that the sum as well as the difference of its prime divisors are primes (2 + 5 = 7 and 5 - 2 = 3)
- 10 is the smallest noncototient, a number that can not be expressed as the difference between any integer and the total of coprimes below it
- 10 is the only number such that its binary representation is the concatenation of two copies of its decimal representation
- A square can be subdivided into 10 acute isosceles triangles, and 10 is the smallest such number
- The sequence of the last letters of the first ten numbers EOEREXNTEN ends in TEN
- 10 is the smallest nontrivial number that is both triangular and tetrahedral
- 10! is the only known factorial which is a product of two non-trivial factorials (10! = 6!7!) outside of the general pattern (n!)!=n!(n!-1)!
- 10 is the smallest number that equals the sum of primes between its smallest and largest prime factors, inclusive

## Rare Properties of 10

A *tetrahedral* number is the number of balls you can put in a triangular pyramid.

This is the space generalization of triangular and square numbers.

## Common Properties of 10

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

The number n is *deficient* if the sum of all its positive divisors except itself is less than n.

Compare with perfect and abundant 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.

One can take the sum of the squares of the digits of a number. Those numbers are *happy* for which iterating this operation eventually leads to 1.

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

If you start with n points on a line, then draw n-1 points above and between, then n-2 above and between them, and so on, you will get a triangle of points. The number of points in this triangle is a *triangle* number.

Compare to square, pentagonal and tetrahedral numbers.