Now I'm asking my first question to understand a specific proof:
Let $n=pq$ and $q,p \in \mathbb{P}$. Then we get $p-1\mid n-1$ and $q-1\mid n-1$, because there are prime integers mod $p$ and mod $q$. Further we get $n-1=pq-1=p(q-1)+p-1$. To this step everything is clear. Now the author says: from $p(q-1)+p-1$ it follows $q-1\mid p-1$ and $p-1\mid q-1$. I don't have a clue how he gets there. Any help is appreciated. Thanks :-)