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This question already has an answer here:

I have to prove that $\sqrt 3$ is irrational. let us assume that $\sqrt 3$ is rational. This means for some distinct integers $p$ and $q$ having no common factor other than 1,

$$\frac{p}{q} = \sqrt3$$

$$\Rightarrow \frac{p^2}{q^2} = 3$$

$$\Rightarrow p^2 = 3 q^2$$

This means that 3 divides $p^2$. This means that 3 divides $p$ (because every factor must appear twice for the square to exist). So we have, $p = 3 r$ for some integer $r$. Extending the argument to $q$, we discover that they have a common factor of 3, which is a contradiction.

Is this proof correct?

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marked as duplicate by Aaron Maroja, Zain Patel, N. F. Taussig, Daniel W. Farlow, user147263 Jul 16 '15 at 20:33

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    $\begingroup$ @AaronMaroja This looks like a proof verification, and not exactly the same proof as that question... $\endgroup$ – miradulo Jul 16 '15 at 18:18
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    $\begingroup$ I would add the step $9r^2 = 3q^2 \implies q^2 = 3r^2$ for clarity. $\endgroup$ – Marconius Jul 16 '15 at 18:18
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    $\begingroup$ @AaronMaroja I don't think that is a duplicate, at least of that question. The thing to prove is the same, but both questions are proof-verifications, and the proofs are quite different. $\endgroup$ – ajotatxe Jul 16 '15 at 18:19
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    $\begingroup$ @DonkeyKong It's just that, there are so many different questions and duplicates over this topic. $\endgroup$ – Aaron Maroja Jul 16 '15 at 18:20
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    $\begingroup$ @brinkle The proof is correct, and rather classical. $\endgroup$ – ajotatxe Jul 16 '15 at 18:21
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Another proof, without arithmetic theorems:

Suppose $x=\sqrt 3$ is rational. Let $q$ be the smallest positive integer such that $qx$ is an integer, and $q'= q(x-1)$. Note it is a natural number since $qx$ is; furthermore $$q'x =qx^2-qx=3q-qx$$ is a natural number.

However, since $1<3<4$, we know $1<x<2$, hence $0<x-1<1$, so that $$0<q'<q$$ which contradicts the minimality of $q$.

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