VN-TST 2015 Problem 3

A positive interger number $k$ is called " $t-m$ " -property if forall positive interger number $a$, there exists a positive integer number $n$ such that ${{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m).$ a) Find all positive integer numbers $k$ which has $t-20$ - property. b) Find smallest positive integer number $k$ which has $t-{{20}^{15}}$ - property.