Question 99
Let $n$ and $k$ be positive integers and suppose that $k$ is odd. Let $r$ be the greatest integer for which $2^r$ divides $n$. Let $h$ be the greatest integer for which $2^h$ divides $1 + k + k^2 + \dots + k^{n-1}$. Prove
(i) if $k \equiv 1 \pmod 4$, then $h = r$;
(ii) if $k \equiv 3 \pmod 4$ and $n$ is odd, then $h = 0$;
(iii) if $k \equiv 3 \pmod 4$ and $n$ is even, then $h = s + r - 1$, where $s$ is the greatest integer for which $2^s$ divides $k+1$.
(i) if $k \equiv 1 \pmod 4$, then $h = r$;
(ii) if $k \equiv 3 \pmod 4$ and $n$ is odd, then $h = 0$;
(iii) if $k \equiv 3 \pmod 4$ and $n$ is even, then $h = s + r - 1$, where $s$ is the greatest integer for which $2^s$ divides $k+1$.