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

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Step 1: Let $\sqrt{5}$ be a rational number. $\sqrt{5} = \frac{p}{q}$, $p$ & $q$ are coprimes & $q 
eq 0$
Step 2: $5q^2 = p^2 \Rightarrow 5$ divides $p^2 \Rightarrow 5$ divides $p$ also. Let $p = 5a$, for some integer $a$
Step 3: $5q^2 = 25a^2 \Rightarrow q^2 = 5a^2 \Rightarrow 5$ divides $q^2 \Rightarrow 5$ divides $q$ also
Step 4: $	herefore 5$ is a common factor of $p$, $q$, which is not possible as $p$, $q$ are coprimes. Hence assumption is wrong. $\sqrt{5}$ is irrational

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