Question

Prove that $5\sqrt{2}$ is irrational.

Answer 1

Let √2 = $\frac{a}{b}$ where a,b are integers b ≠ 0
We also let that $\frac{a}{b}$ is written in the simplest form.
⇒ √2 = $\frac{a}{b}$

⇒ 2 = $\frac{a^2}{b^2}$

⇒ 2$b^2=a^2$
∴ $2b^2$ is divisible by 2
⇒ $a^2$ is divisible by 2
⇒ a is divisible by 2
∴ Let a = 2c
$a^2$ = 4 $c^2$

⇒ 2 $b^2$ = 4 $c^2$ ⇒ $b^2$2 = 2 $c^2$
∴ 2$c^2$ is divisible by 2
∴ $b^2$ is divisible by 2
∴ b is divisible by 2
∴a are b are divisible by 2 .
This contradicts our notion that a/b is written in the simplest form as our notion is wrong
∴ √2 is irrational number.
If we multiply 5 in this so $5\sqrt{2}$ is also irrational.