TITLE: If a divisor of $pq-1$ divides the LCM of $p-1$ and $q-1$, then it also divides the GCD of these two numbers
QUESTION [8 upvotes]: Suppose that $p,q$ are distinct odd primes. Suppose an integer $k$ divides $pq-1$ and also $k|\operatorname{lcm}(p-1,q-1)$. Show that $k|\operatorname{gcd}(p-1,q-1)$.

I've spent ages looking at this problem and very little to show.  Surely I would want to use the hypothesis that p and q are not equal.  This would mean their gcd is 1.  How to use this assumption?

REPLY [2 votes]: This is true for arbitrary integers $p$ and $q$, not just distinct odd primes.
It suffices to show that every prime power dividing both $\mathrm{lcm}(p-1,q-1)$ and $pq-1$  must also divide $p-1$ and $q-1$.
Suppose $\ell^n$ is a prime power dividing $\mathrm{lcm}(p-1,q-1)$ and $pq-1$. The first divisibility implies $p\equiv 1$ or $q\equiv 1\mod\ell^n$. The second divisibility means $p\equiv q^{-1}\mod\ell^n$, so whichever of $p$ or $q$ is congruent to $1$, the other is also. This shows $\ell^n|p-1$ and $\ell^n|q-1$.