Question 665192
Don't be fooled by the "greater than 5 billion" in the problem statement.  The only thing that matters is the ones digit.  
 
Think of it this way:  Every integer can be expressed as the sum of a multiple of 10 and a single digit (e.g. for any integer x, x = 10k + i where 0 <= i <= 9).  So, 
{{{x^5 = (10k + i)^5}}}
{{{x^5 = (10k)^5 + 5(10k)^4*i + 10(10k)^3*i^2 + 10(10k)^2*i^3 + 5(10k)*i^4 + i^5}}}
Note that every term is multiple of 10 except {{{i^5}}}, so the ones digit of {{{x^5}}} is the same as the ones digit of {{{i^5}}}.  That is 
{{{x^5}}} mod 10 = {{{i^5}}} mod 10}}}
  
So any 10 consecutive integers will have ones digits of 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 (not necessarily in that order).  Raising each of those integers to the 5th power will result in numbers with ones digits equal to the ones digits of {{{0^5, 1^5, 2^5, 3^5, 4^5, 5^5, 6^5, 7^5, 8^5, and 9^5}}}, again, not necessarily in that order.  
 
So all we need to know is:
{{{0^5}}} mod 10 = 0 mod 10 = 0
{{{1^5}}} mod 10 = 1 mod 10 = 1
{{{2^5}}} mod 10 = 32 mod 10 = 2
{{{3^5}}} mod 10 = 243 mod 10 = 3
{{{4^5}}} mod 10 = 1024 mod 10 = 4
{{{5^5}}} mod 10 = 3125 mod 10 = 5
{{{6^5}}} mod 10 = 7776 mod 10 = 6
{{{7^5}}} mod 10 = 16807 mod 10 = 7
{{{8^5}}} mod 10 = 32768 mod 10 = 8
{{{9^5}}} mod 10 = 59049 mod 10 = 9
 
So the ones digit we are looking for is 
(0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) mod 10 = 45 mod 10 = 5
 
The ones digit of the sum is 5.