Prove by contradiction that there is an infinite number of prime numbers.

The 'by contradiction' tells us we need to assume the opposite to begin with: 1) Let's assume there is a finite number of prime numbers2) Let P be the largest prime number (the last one) 3) if we multiply all the prime numbers up to and including P: 2x3x5x7...xP=q (the multiple of all prime number up to and including P)4) consider q+1 5) Will it be divisible by any prime P or less? no, as q is divisible by those and q+1 is only 1 more.6) So this means that either q+1 is Prime, or it has a prime factor larger than P.7) But P is the largest prime factor - this is a contradiction as there must exist a prime larger than PHence there is an infinite number of prime numbers