# Finding all Divisors (or Factors) of Whole Numbers

Any whole number can be represented as a product of 2 or more integers - these are known as the *divisors* or *factors* of this number.

A **proper divisor** is a divisor other than the number itself (but including 1).
The technique used to determine all factors is the cartesian product of the sets of powers of the prime factors.

e.g. To determine all factors of 180, we first factorize 180 into its prime factors:

180 = 2 x 2 x 3 x 3 x 5

We now collect these into the powers of the respective primes

{2⁰, 2¹, 2²}

{3⁰, 3¹, 3²}

{5⁰, 5¹}

Note that the power of 0 has been added (any number to the power of zero is 1).

Now, the cartesian product of these sets is determined

{1, 2, 4}

{1, 3, 9}

{1, 5}

1x1x1, 2x1x1, 4x1x1, 1x3x1, 1x9x1, .... 4 x 9 x 5

1 , 2 , 3 , 4 , 5 , 6 , 9 , 10 , 12 , 15 , 18 , 20 , 30 , 36 , 45 , 60 , 90 , 180