Is it true that square root of any positive integer greater than one, which is not a perfect square, is an irrational number?
This question was raised during a discussion with first year degree students. The motivation was the proof showing the irrationality of $\sqrt{2}$. The discussion took us to the following 'one line' proof: If $n$ is such a number having the rational expression $\sqrt{n}=\frac{a}{b}, \gcd(a,b)=1$, then by the fundamental theorem of arithmetic (unique prime factorization) $b=1$ and $n$ is a square.
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