Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 1, Problem 2

Problem

Prove that there is no rational number whose square is 12.


Answer

Suppose there exists a rational number $p$ that satisfies $p^2=12$. Since $p$ is a rational number, there are some integers $m,n$ that are not both multiples of 3 and $p=m/n$. Then, we have $12n^2=m^2$. This shows that $m$ is a multiple of 3. Then $n$ also has to be a multiple of 3 and this is a contradiction.

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