So pretend that the five numbers are a, b, c, d and e. If you take into account that each of the numbers is used four times whilst making the pairs and that each of the pair's sums add up to 72, you would divide 72 by 4 which is 18. That means that a add b add c add d add e is 18.
To find out what a, b and c are you'll have to see that a+b=0, a+c=2 and b+c=4. If you use trial and improvement you will get the answer that a=-1, b=1, c=3.
Now you have to find out what d and e are there is a simple way of working this out. If you take the fact that all the five pair numbers add up to 18 and that a+b+c=3, you should be able to work out that d+e= 15. So if again you use trial and improvement you will eventually get the correct pair of numbers: d=5, e=10.
Now you should have the five numbers which fit the problem, all of the pairs will be one of the numbers ( i.e. 0,2,4,4,6 etc.), none will be rogue numbers as in they wouldn't have a pair which would not add up to a number on the list. So if you follow all the rules above you should have got the answer: a=-1, b=1, c=3, d=5, e=10.