Proof square root of 4 is not irrational.
All you can prove with this strategy is that $m=2n$, but the HCF is $1$ if we take $m=2,\,n=1$. By contrast, the analogous treatment of $\sqrt{2}$ shows $m,\,n$ must both be even.
All you can prove with this strategy is that $m=2n$, but the HCF is $1$ if we take $m=2,\,n=1$. By contrast, the analogous treatment of $\sqrt{2}$ shows $m,\,n$ must both be even.