Question 3
A positive integer $k$ has property $T(m)$ if, for every positive integer $a$, there is a positive integer $n$ such that
$$1^k+2^k+\cdots+n^k\equiv a\pmod m.$$
(a) Find all positive integers $k$ with property $T(20)$.
(b) Find the smallest positive integer $k$ with property $T(20^{15})$.
$$1^k+2^k+\cdots+n^k\equiv a\pmod m.$$
(a) Find all positive integers $k$ with property $T(20)$.
(b) Find the smallest positive integer $k$ with property $T(20^{15})$.