Let $N$ be a positive integer. Consider the relation $\circledast$ among pairs of integers $r, s \in \mathbb{Z}$ defined as $r \circledast s$ when $r-s$ is an integer multiple of $N$. Prove that $\circledast$ is an equivalence relation and identify the elements of the quotient space $Q=\mathbb{Z} / \circledast$. Prove that the composition law among equivalence classes $\odot$ defined as $[a] \odot[b]=[a+b]$ endows $Q$ with group structure, and identify to which group $(Q, \odot)$ is isomorphic.