how to determine if a number is prime number??

actually my way to determine if a number is a prime number is by trial and error test.

For example given a number 91, to see whether it is a prime number or not i try to use prime number like 3,5,7,11,13,17 etc to see if it can divide 91 without reminder.

but is there any explanation to why it is? or is there any other more systematic method? Because i need to explain to my students.

By the way, i would like to ask what is the mathematical significance (or simply why we should do) "prime factorization"?