Divisibility, the property of an integer being a multiple of another integer, and many other related concepts such as primality, Greatest Common Divisor (GCD), Least Common Multiple (LCM) etc. are central to the learning of Number Sense and form the basis for working with fractions and many other higher level math topics. The notion of divisibility and related operations are usually introduced in elementary schools over a number of years, but are rarely presented as a collection of related ideas. Also, they tend to skip certain kind of reasoning in favor of rote memorization. For example, kids might learn that if the sum of digits of a number is divisible by 3 then the number is also divisible by 3, but may not know why this rule works.
Advanced math texts focus more on formal and structured treatment full of lemmas, theorems and formal proofs under the general subject of Number Theory.
So I planned to start this year's MATHCOUNTS club with an overview of Divisibility. As there is rarely enough time in the class to go over all the topics in sufficient detail, I thought it will be good to point them to relevant stuff on the Web. However, my search drew blank. Wikipedia articles are good but too advanced and disjointed for an average middle school student. Relevant articles at other sites also didn't stand up to my expectations.
So I resolved to roll my own tutorial: Understanding Divisibility. It takes somewhat unconventional approach to introduce ideas and provide insight on mathematical reasoning by stating them in plain English and avoiding too much use of notations that may be a put-off for the younger crowd, such as modulo arithmetic for explaining rules of divisibility.
The current version at the time of this post is still rough at edges but I plan to polish it over time based on feedback here and discussion in the class.