Another cute proof: square root of 2 is irrational.

Reference: Elementary Number Theory, David M. Burton, Sixth Edition, Tata McGraw-Hill.

(We are all aware of the proof we learn in high school that $\sqrt{2}$ is irrational. (due Pythagoras)). But, there is an interesting variation of that proof.

Let $\sqrt{2}=\frac{a}{b}$ with $gcd(a,b)=1$ , there must exist integers r and s such that $ar+bs=1$ . As a result, $\sqrt{2}=\sqrt{2}(ar+bs)=(\sqrt{2}a)r+(\sqrt{2}b)s=2br+2bs$ . This representation leads us to conclude that $\sqrt{2}$ is an integer, an obvious impossibility. QED.