Question

Which methods of factoring do you use to factor completely?

Answer 1

You (try to) divide your number by successive primes `2,3,5,7,11...`

Example :
Say you want to factor `756`
Even number, divide by 2: `756=2*378`
Even, divide by 2: `756=2*2*189`
Odd, so try 3: `756=2*2*3*63`
And again: `756=2*2*3*3*21`
And again: `756=2*2*3*3*3*7`
7 is a prime, so you're finished.

You write the result as `756=2^2*3^3*7`

Remark :
You never have to try a prime that is larger than the square root of the number left, because if that would divide evenly, the result is a smaller number, and you would have already tried that.