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I looked at the proof of $\sqrt 2$ is irrational. In first step we assume that $\sqrt 2$ is rational. Then we say it should be written as $\frac ab$ if it's rational.After that we assume gcd(a,b)=1 and end of the calculations we conclude that a and b are even numbers.So there is a common factor but we assumed that gcd(a,b)=1 so it's a contradiction , $\sqrt 2$ must be irrational.The thing that I don't understand why we assume that gcd(a,b)=1 ? We assume that $\sqrt 2$ is rational it's okay but why we need to assume gcd(a,b)=1.I don't think it's the need of being rational ?