What is the probability that the digit sum of a randomly chosen integer between 0000 and 9999 is divisible by 5? If I have a randomly selected integer between 0000 and 9999, what is the probability that the digit sum of that number is divisible by 5? 
[E.g. 1234 = 1 + 2 + 3 + 4 = 10]
I've started off with knowing that I have 2 options for the last integer, but I'm not sure where to go from there.

Javi is almost right! Why focus on the last digit?
For example the digit sum of 104 is divisible by 5. So we have to check the possiblities. The smallest sum is zero and the biggest possible sum having four digits is four times nine = 36, as Javi just explained correctly.
Now let's look how many of the possible digit sums are divisible by 5:
0, 5, 10, 15, 20, 25, 30 and finally 35.
So one could assume that P is 8/36 $\approx$ 0,22. But thats not true...
Looking at the digit sums for each decade one will see that there are always only two sums divisable by 5. So the value of 0,2 or 20 % is correct.
