Project Euler Problem 1

I recently attempted many of the simpler Project Euler problems, and decided it would be good to write down my thinking process.  If you don't know, Project Euler is a collection of programming problems with a heavy emphasis on mathematics.  They highly discourage sharing the answers since they want you to solve the problems by yourself, but since answers are already scattered all over the Internet it's up to you to challenge yourself and avoid spoilers.  Consider yourself warned - this post contains solutions and discussions thereof.

Problem statement:  Add all the natural numbers below one thousand that are multiples of 3 or 5.