So you have a number, say 153, and want to know whether it is divisible by 3. Perhaps this is a money problem, or a prime factor decomposition, or something else entirely. Who knows, but you want to divide something by 3 none the less.
How do you know whether its worth getting your pencil and paper out, ready for the long division? Or whether it's worth reaching in your bag for the calculator. You can work out whether your efforts are wasted pretty easily.
Take the digits, add them up, 1+5+3 = 9
If the results is divisible by 3, so is the original number. In this case, 153 is divisible, so doing that long division is a good use of time.