Question 170
a) Find all positive integers $k$ with the property $T(20)$, where a positive integer $k$ has the property $T(m)$ if for any positive integer $a$, there exists a positive integer $n$ such that
$$
1^k + 2^k + \dots + n^k \equiv a \pmod{m}.
$$
b) Find the smallest positive integer $k$ with the property $T(20^{15})$.
$$
1^k + 2^k + \dots + n^k \equiv a \pmod{m}.
$$
b) Find the smallest positive integer $k$ with the property $T(20^{15})$.