Give an example of how prime factorization could be used in the real world.

When you look behind the Idea of what prime factorization is, you will see that it is about breaking things down to their most basic parts. If you pay attention to the world, and your mind, when you learn math, you will get great life-lessons from learning about things like how we sort the world. Sorting numbers is great lesson in sorting reality. 
What do I mean by this? 

OK, an example: let's say you are trying to understand the way a good government works. You decide, (or discover) what the basic, inalienable parts are. That means things which you can never, ever break if you want to build a good government. I would say the first 10 amendments of the Constitution of the United States (also known as "The Bill of Rights" are a good example. You cannot break those laws while making other laws, and still have a good government. They are the PRIME FACTORS in making laws. 

The american colonies would have been ruled by a king.