From the introduction: Most students are aware of the fact that a number $N$ is divisible by nine exactly when the sum of the digits of $N$ is a multiple of nine. This fact gives a shortcut that allows one to determine if nine is a divisor of $N$ without performing a division (for large numbers it may be necessary to apply this fact more than once). Far fewer students know that a similar shortcut applies to the number eleven. In this note we will consider various shortcuts that can be used to determine if a given value divides a number $N$. In most cases these techniques are significantly faster than division. For completeness this note will include some techniques that are fairly â€śobvious", but others of these shortcuts should be new to most students. This number analysis may be of interest to college mathematics groups or motivated liberal arts students.