# 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 125

- 125 is the smallest cube which is also a sum of distinct non-zero squares: 125 = 5
^{3} = 11^{2} + 2^{2}
- 125 is the smallest hypotenuse for which there exist exactly 3 distinct Pythagorean triangles
- 125 is the smallest number n such that n
^{n} contains a pandigital substring
- 125 is the smallest cube with the same set of digits as another cube (512)
- 125 is the smallest cube that is sandwiched between two non-squarefree numbers: 124 = 2
^{2} * 31, 125 = 5^{3}, 126 = 2 * 3^{2} * 7 (submitted by Massimo Koffler)

## Rare Properties of 125

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

## Common Properties of 125

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}.