Find $a,b\in\Bbb Z$ where $S_k=1^k+2^k...+p^k$ for $(p-1)|k$ with $S_k\equiv a\equiv b\pmod{p}$ & $0\le a<p$ and $-p< b\le 0$.

Question

The question goes as follows:

Let $S_k$ be the sum of $1^k + 2^k ... + p^k$, where $p$ is an odd prime and $k$ is a multiple of $p-1.$

Find 2 integers "$a$" and "$b$" where:

• $S_k \equiv a \equiv b \pmod{p}$
• and $0 \le a < p$ and $-p < b \le 0$.

My logic is to start by using Fermat's Little Thm to simplify things, but i don't really know how to start that.