$n-1 = pq-1 \equiv q-1\pmod{p-1}$ where $n\in\mathbb{N}$ such that $n=pq$ for two distinct large primes $p$ and $q$.

Question

Let $n\in\mathbb{N}$ such that $n=pq$ for two distinct large primes $p$ and $q$. My lecturer simply states that

$$n-1 = pq-1 \equiv q-1\pmod{p-1}$$

without any justification and I can't see how this is true given the information we know. Could someone explain?

Answer 1

It is just $pq-1 = q(p-1)+q-1$.