Find your own links for this if you want - I don't recall where I picked them up.
Any idiot can figure out whether a number is divisible by 2 - is it even? Put another way, if the last digit is divisible by 2, so is the whole number. For four, check the last 2 digits, for 8 check the last 3, ad nauseam. It works because 10 is divisible by 2.
What about 3? Add up the digits in the number. If the sum is divisible by 3, so is the original number. Is the sum of the digits too big? Repeat the process.
We covered 4. The rule for 5 is easy. 6 works if the number is even and divisible by 3. 9 works like 3, except the sum must be 9. 1 and 10 will be left as exercises.
What about 7? Beats me. But the decimals have an interesting property - they consist of a string of 142857. For 1/7, start with the 1 - .142857.... Start with the next smallest in the group for each successive increment of the numerator, thus 2/7=.285714..., 3/7=.428571..., and so forth.
Alright, I'm a geek. But this stuff can come in handy sometimes, even in a world where people don't even have to count out change anymore.
Wanna get hard-core? Try this. As I recall, the inventor was a prisoner in a Nazi concentration camp, and worked out these methods to keep himself sane.