So the question is to show that every residue class $\pmod{2^a}$ can be written as $\pm 5^r$ for some $r$.
The hint is to first show that:
For $a \ge 3$, and $H$ the multiplicative subgroup of $(Z/2^aZ)^*$ generated by $5\pmod{2^a}$ show that $-1 \notin H$
This is a homework question, so I'm not looking for an answer... but at this point I don't even know how to begin showing this. I really was hoping for no group theory to be in this course..