Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 1I / i

Suppose the prime numbers were the finite list $p_1,\dots ,p_r$. The integer $N=p_1\cdots p_r+1>1$ has a prime factor, by repeatedly factoring any composite factor until a prime is reached. But division by any $p_j$ leaves remainder one, so that prime factor is absent from the list. This contradiction proves there are infinitely many prime numbers. The argument does not require $N$ itself to be prime.