Is it possible to find a set of 5 integers, such that the 10 sums each have a different last digit? Full question: 

If we are given 5 integers, there are $\binom{5}{2} = 10$ different ways to find the sum of 2 of the integers.
  Is it possible to find a set of 5 integers, such that the 10 sums each have a different last digit?

Am I trying to prove that 5 integers can be found, such that the 10 sums each have a different last digit, or am I being asked to find 5 such integers?

Let $o$ be the number of odd integers in the set. Then you can form $o(5-o)\ne5$ odd sums. Therefore a set of the desired kind does not exist.
