Prove that 5 2 is an irrational number. - Vidyadip

Question

Prove that $5\sqrt2$ is an irrational number.

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Answer

Let $5\sqrt2$ is a rational number.
$\therefore5\sqrt2=\frac{\text{p}}{\text{q}},$ where p and q are some integers and HCF $(p, q) = 1 ...(1)$
$\Rightarrow5\sqrt2\text{q}=\text{p}$
$\Rightarrow\big(5\sqrt2\text{q}\big)^2=\text{p}^2$
$\Rightarrow2(25\text{q}^2)=\text{p}^2$
$\Rightarrow p^2$ is divisible by $2$
$\Rightarrow p$ is divisible by $2 ...(2)$
Let $p = 2m$, where m is some integer.
$\Rightarrow5\sqrt2\text{q}=\text{2m}$
$\Rightarrow\big(5\sqrt2\text{q}\big)^2=\text{2m}^2$
$\Rightarrow2(25\text{q}^2)=\text{4m}^2$
$\Rightarrow\text{25q}^2=\text{2m}^2$
$\Rightarrow q^2$ is divisible by $2$
$\Rightarrow q$ is divisible by $2 ...(3)$
From $(2)$ and $(3)$, $2$ is a common factor of both $p$ and $q$, which contradicts $(1)$.
Hence, our assumption is wrong.
Thus, $5\sqrt2$ is irrational.

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