Let $k=b_nb_{n-1}\ldots b_3b_2b_1b_0$ be the binary representation of an odd positive integer.
Prove:
If $k\equiv 1 \mod 4$ then $b_1=0$.
If $k\equiv 3 \mod 4$ then $b_1=1$.
I think that to prove the above I need to use $k=1 +4q$ or $k=3 +4q$ for $q\geq 0$. Any suggestions?