# 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 27

- 27 is the only number which is thrice the sum of its digits
- 27 is the first composite number not divisible by any of its digits
- 27 is the largest number that is the sum of the digits of its cube
- 27 is the only 2-digit number in which the sum of digits is equal to the sum of prime factors (27 = 3 * 3 * 3 and 2 + 7 = 3 + 3 + 3 = 9)
- A 10,000-day-old person is 27 years old
- 27 is the smallest cube out of three known with only prime digits (the other cubes are 3375 and 2775577757352755375573357273)
- A web page about 27: The Mystery of the number 27
- 27 is the smallest evil cube
- 27 is the largest number that equals the sum of numbers between the first and the second digits inclusive
- 27 is the smallest number that have composite sum of digits and prime sum of squares of digits
- 27 is the smallest number n where n and n+1 are both products of 3 primes
- 27 is the smallest number whose reciprocal has period 3

## Rare Properties of 27

The number n is a *cube* if it is the cube of an integer.

## Common Properties of 27

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.

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

Guess what odious numbers are.

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.

An integer n is *powerful* if for every prime p dividing n, p^{2} also divides n.

How much power? They all can be written as a^{2} b^{3}.

A composite number is called a *Smith* number if the sum of its digits equals the sum of all the digits appearing in its prime divisors (counting multiplicity).

In 1984, when Albert Wilansky called his brother-in-law, named Smith, he noticed that the phone number possesses the property described here. Are they called joke numbers, because they were named after an innocent unsuspecting brother-in-law :-) ?