(i) Let $N$ be an integer $\geqslant 2$. Define the addition and multiplication on the set of congruence classes modulo $N$.

(ii) Let an integer $M \geqslant 1$ have expansion to the base 10 given by $a_{s} \ldots a_{0}$. Prove that 11 divides $M$ if and only if $\sum_{i=0}^{s}(-1)^{i} a_{i}$ is divisible by 11 .