Question

Prove that √5 is an irrational number.

Answer 1

Suppose that √5 is rational

Then

So let p = 5c 

Substitute p = 5c in (1) we get 

(5c)² = 5q² ⇒ 25c² = 5q²

c² = 5q²/25 = q²/5

⇒ 5 is a factor of q also 

So 5 is a factor of p and q which is a contradiction. 

⇒ √5 is not a rational number 

⇒ √5 is an irrational number