Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 1H / i

By the Korselt criterion, the prime factors are distinct. There cannot be just one prime factor, since a Carmichael number is composite. If there were exactly two, write $n=pq$ with $p<q$. Then $q-1\mid pq-1=p(q-1)+(p-1)$, so $q-1\mid p-1$, impossible because $0<p-1<q-1$. Hence at least three distinct prime factors are necessary.