# 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 167

- 167 is the smallest prime in which the digits can be partitioned in three non-zero parts such that the sum of the first two parts equals the third one
- 167 is the smallest number whose 4th power begins with 4 identical digits: 167
^{4} = 777796321
- 167 is the smallest multidigit prime such that the product of its digits is equal to the number of digits times the sum of digits

## Common Properties of 167

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.

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 *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.