Prove the fractions aren't integers 
Prove that if $p$ and $q$ are distinct primes then $\dfrac{pq-1}{(p-1)(q-1)}$ is never an integer. Is it similarly true that if $p,q,r$ are distinct primes then $\dfrac{pqr-1}{(p-1)(q-1)(r-1)}$ is also never an integer?

I think using a modular arithmetic argument here would help. In other words, we must have for the first fraction $pq-1 \equiv 0 \pmod{(p-1)(q-1)}$. Then I am unsure how to proceed next since we can't really use the Chinese Remainder Theorem. 

Suppose, for the sake of contradiction, such distinct $p$ and $q$ exist.
First of all observe that the statement implies that $p-1|pq-1$. So,
$$p-1|pq-1-q(p-1) \implies p-1|q-1$$
Similarly we get,
$$q-1|p-1$$
These observations imply that $p-1 = q-1$. This implies that $p = q$. Contradiction. They aren't distinct.
