# 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 227

- 227 is the number of composite days in a year
- 227 is the smallest three-digit prime that is changed into a composite number if any digit is deleted
- 227 is the smallest prime such its sum of digits is prime and the sum of digits of its index is the next prime after the sum of digits of the number itself (227 is the 49th prime, while the sum of digits is 13, which is the next prime after 11, the sum of digits of 227)
- 227 is the smallest number that remains prime after incrementing each digit by the same even number without carry (numbers 227, 449, 661, and 5 are all prime)

## Common Properties of 227

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 *odd* if it is not divisible by 2.

Numbers that are not odd are even. Compare with another pair -- evil and odious numbers.

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

Guess what evil numbers are.

A *prime* is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

Prime numbers are opposite to composite numbers.

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

A prime number is called a *twin* prime if there exists another prime number differing from it by 2.