Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 11G / c

The converted source omits $r$ from the product; the PDF asks about $N=pqr$. Fix the odd prime $r$, and suppose $pqr$ is Carmichael. Its three prime factors are distinct by the Korselt criterion. Relabel $p,q$ so that $p<q$.

The divisibility conditions includeWritefor a positive integer $b$. Since $p<q$,so $1\le b<r$. Moreover,Multiplying the second divisibility by $b$ and reducing modulo $p-1$ givesThe positive integer on the right is at most $(r-1)(2r-1)$. ConsequentlyThere are only finitely many possible primes $p$ and integers $b\in \{1,\dots ,r-1\}$, and each pair determines at most one $q=(pr+b-1)/b$. Therefore