Prime Factorization

To find all the factors, of any number, you will need to find a set of prime factors. This works for all numbers. The set of prime factors will Always be the same

Finding Sets Example

For the number 21, I am to find all the prime factors of it The prime factors are: 2,3 and 3 this works if I chose other prime factors in different steps too. 3,2,2. same set.

Finding the factors

{2,3,3} is the set. 1 is also in the set Mutliply every number in the set with every other number all possible times to get all factors. 1 * 1 = 1 2 * 1 = 2 3 * 1 = 3 2 * 3 = 6 3 * 3 = 9 2 * 3 \ * 3 = 18 factors are: 1,2,3,6,9,18

Finding Perfect Square Factors

Say you have a number 3600, you want to get all perfect square factors. The prime factorization of 3600 is $2^4\ast 3^2\ast 5^2$ A perfect square occurs when the degrees of the terms are even. So $2^0\ast 3^2\ast 5^2$ or $2^4\ast 3^2\ast 5^0$ as examples, there are a lot more.