2011/028) to prove that sqrt(2) is irrational

before proving it we need to prove a lemma

lemma: if 3 / (a^2+ b^2) then 3 | a and 3| b

proof:

we have a mod 3 = 0 or 1 or 2

so a^2 mod 3 = 0 or 1
and b^2 mod 3 = 0 or 1

so a^2 + b^2 mod 3 = 0 if a mod 3 = 0 and b mod 3 = 0
or 1 or 2 otherwise

hence lemma is proved

now for the proof of sqrt(2) is irrational

let sqrt(2) = a/b where GCD(a,b) = 1 ( we can always reduce it in this form)

so a^2 = 2b^2 or a^2 + b^2 = 3b^2

so 3 / (a^2+b^2) or hence as per above lemma 3 | a and 3 | b

so GCD(a,b) is not 1 so a contradiction

so sqrt(2) is irrational