**Lemma:** Suppose $a, b, c \in \mathbb{N}$. If $a|b \text{ and } a|c \text{ then } a|(b+c)$ **Proof:** $\begin{gather} \text{If } a|b \text{ then } \exists d,e \in \mathbb{N} \text{ s.t. } b=ad\text{ and } c=ae \\ \text{Then } b+c = ad+ae = a(d+e) \text{ so } a|(b+c) \end{gather}$