A test to determine whether a number is divisible by 29
Remove the last digit of the number, add three times that digit to the truncated number, and keep repeating this procedure until a number less than 30 is obtained. The original number is divisible by 29 if and only if the final number is 29.
Why it works
If the number is question is x and the last digit of x is d, then removing the last digit corresponds to subracting d and dividing by 10, i.e. we have (x - d)/10 as our truncated number. If we then add three times d to this we obtain (x - d)/10 + 3d = (x + 29d)/10 . Note that this number is a multiple of 29 if and only if x is a multiple of 29. Also, if x > 29 then (x + 29d)/10 < x. One step of our procedure therefore gives us a smaller number which is a multiple of 29 if and only if our original is a multiple of 29. Repeating the procedure, we must eventually get a number less than 30. Since divisibility or non-divisibility by 29 is preserved throughout the process, if our final number is 29, the original number must have been a multiple of 29; if our final number is not 29, the original number could not have been a multiple of 29.
A test to determine whether a number is divisible by 21
Remove the last digit of the number, subtract twice that digit from the truncated number, and keep repeating this procedure until a number less than 21 is obtained. The original number is divisible by 21 if and only if the final number is 0.
Why this test works
It's similar to the argument above. Try to find other generalizations.