Proof

Question 1

Prove by contradiction that there are infinitely many prime numbers

Answer 1

• Assume there are a finite number of primes, p1, p2, p3,…,pn
• Consider the number: N=p1p2p3…pn+1
• if you divide N by any of the primes p1…pn there is a remainder of 1
• none of the primes are a factor of N
• either N is prime or it has a prime factor which is not in the list
• this is a contradiction to the assumption
• therefore there are infinitely many primes

Question 2

Prove by contradictions that [2 is an irrational number

Answer 2

Assume [2 is rational
Therefore [2=a/b so that a and b are co prime
So 2=a2/b2 hence 2b2=a2this means a2 and a is even so a=2p
A2=4p2so 2b2=4p2 and b2=2p2
So b2 and b is even hence a and b are even
This is a contraction to the assumption that a and b are co prime therefore [2 is irrational