number theory - Prove that for any integer $k>1$ and any positive integer $n$, there exist $n$ consecutive odd integers whose sum is $n^k$

Found these problems in a problem book and got stuck. The book doesn't have solutions to I've come here for help.

(1) Prove that for any integer $k>1$ and any positive integer $n$, there exist $n$ consecutive odd integers whose sum is $n^k$.

(2) Let $n$ be a positive integer and $m$ any integer of the same parity as $n$. Prove that there exist $n$ consecutive odd integers whose sum is $mn$.