Prove that $\sqrt{5}$ is irrational.

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Solutions: Let us assume, that $\sqrt{5}$ is rational number.
i.e. $\sqrt{5}=x / y$ (where, $x$ and $y$ are co-primes) $y \sqrt{5}=x$
Squaring both the sides, we get,

$(y \sqrt{5})^{2}=x^{2}$

$\Rightarrow 5 y^{2}=x^{2} \ldots$

Thus, $x^{2}$ is divisible by 5 , so $x$ is also divisible by 5 .
Let us say, $x=5 k$ , for some value of $k$ and substituting the value of $x$ in equation (1), we get, $5 y^{2}=(5 k)^{2}$ $\Rightarrow y^{2}=5 k^{2}$
$\Rightarrow y^{2}=5 k^{2}$ (a $y^{2}$ is divisible by 5 it means $y$ is divisible by 5 .
Clearly, $\mathrm{x}$ and $\mathrm{y}$ are not co-primes. Thus, our assumption about $\sqrt{5}$ is rational is incorrect.
Hence, $\sqrt{5}$ is irrational number.

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