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How can one prove that the square root of a prime number (e.g. $ \sqrt 61$) is irrational.
First we need to prove that $61$ is prime. This can be done by simply showing that $$ 49 < 61 < 64$$ and so $$ 7 < \sqrt 61 < 8$$ then the only possible prime factors of $61$ are $2 , 3, 5, 7$. Then by contradictions and DIC we can show that $61$ is prime.
Now how can we show that $\sqrt 61$ is irrational?