# 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 254

- 254 is the smallest composite number all of whose divisors (except 1) contain the digit 2
- 254 is the smallest number that is a concatenation of two numbers i and j such that n equals the sum of i-th and j-th prime numbers (2-nd and 54-th prime numbers are 3 and 251, which sum up to 254)

## Common Properties of 254

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 n-th *lazy caterer* number is the maximum number of pieces a (circular) pizza can be cut into with n (straight-line) cuts.

Unlike the situation with cake, everybody gets the toppings.

The number n is *odious* if it has an odd number of 1's in its binary expansion.

Guess what evil numbers are.

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