Factors of a positive integer n are all the positive integers that divide n with remainder equal to zero.

For example 2, 3, 4 and 6 are factors of 12 since

12 / 2 = 6, 12 / 3 = 4, 12 / 4 = 3 and 12 / 6 = 2.

These factors are also present when 12 is written as a product of factors (factoring).

For example 12 = 6 . 2, 12 = 4 . 3 ...

The fundamental theorem of arithmetic states that there is only one way that a given positive integer can be represented as a product of one or more primes numbers.

A Prime number n is a positive integer greater than 1 that has only 1 and n (itself) as positive integer divisors.

Below is a list of the first prime numbers

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,...

How to find prime Factors of a positive integer?

We start by dividing the given number by 2 if possible, then divide the result by 2 if possible; if not divide by the next prime number 3 and continue until the quotient is a prime number

__Example 1:__ Factor 4 into prime numbers

Solution

4 = 2 . 2 = 2^{ 2}
__Example 2:__ Factor 20 into prime numbers

Solution

step 1. 20 = 2 . 10

step 2. 20 = 2 . 2 . 5

20 = 2 . 2 . 5 = 2^{ 2} . 5
__Example 3:__ Factor 100 into prime numbers

Solution

step 1. 100 = 2 . 50

step2. 100 = 2 . 2 . 25

step3. 100 = 2 . 2 . 5 . 5

100 = 2 . 2 . 5 . 5 = 2^{ 2} . 5^{ 2}
__Example 5:__ Factor 1020 into prime numbers

Solution

step 1. 1020 = 2 . 510

step 2. 1020 = 2 . 2 . 255

step 3. 1020 = 2 . 2 . 3 . 85

step 4. 1020 = 2 . 2 . 3 . 5 . 17

1020 = 2 . 2 . 3 . 5 . 17 = 2^{ 2} . 3 . 5 . 17
__Example 6:__ Factor 634 into prime numbers

Solution

step 1. 634 = 2 . 317

634 = 2 . 317
__Example 7:__ Factor 720 into prime numbers

Solution

step 1. 720 = 2 . 360

step 2. 720 = 2 . 2 . 180

step 3. 720 = 2 . 2 . 2 . 90

step 4. 720 = 2 . 2 . 2 . 2 . 45

step 5. 720 = 2 . 2 . 2 . 2 . 3 . 15

step 6. 720 = 2 . 2 . 2 . 2 . 3 . 3 . 5

720 = 2^{ 4} . 3^{ 2} . 5
More on numbers and Fractions.