For a ring $R= (a/b|a,b \in \mathbb{Z}, b \text{ odd})$, define the ideals $I_k$ (where $I_k$ only applies to non-negative integers $k$), as $I_k = 2^kR=(2^k a/b | a, b \in \mathbb{Z}, b \text{ odd})$, show that 0,1,2,3,4,5,6,7 represent all congruence classes mod $I_3$.
I would really appreciate it if someone could explain this to me in detail and provide a rigorous proof. I also don't understand and would like to know that for any ring R, not just necessarily this one, how would one go about checking the congruence classes$\mod I_3$? Many thanks.