1^x+2^x+3^x+4^x+5^x

Ending digit patterns for  all five digits raised to any positive powers:

Power 1   2   3  4 

1 =      1   1   1  1 

2 =      2   4   8  6 

3 =      3   9   7  1 

4 =      4   6   4  6  

5 =      5   5   5  5 

And these patterns repeat  for every group of 4 powers, i.e., [5 - 8], [9 - 12], etc.

Notice that the ending digit sum for the odd powers end in 5

So the ending  digit sums for all odd powers end in 5......and all such numbers are multiples of 5