Given a finite list of prime numbers \(p_{1}, \ldots, p_{N}\), let \(M=\) \(p_{1} \cdot p_{2} \cdots p_{N}+1\). Show that \(M\) is not divisible by any of the primes \(p_{1}, \ldots, p_{N}\). Use this and the fact that every number has a prime factorization to prove that there exist infinitely many prime numbers. This argument was advanced by Euclid in The Elements.