Prove that root 5is a irrational …

Let us prove {tex}\sqrt 5 {/tex} irrational by contradiction.
Let us suppose that {tex}\sqrt 5 {/tex} is rational. It means that we have co-prime integers a and b (b ≠ 0)
Such that {tex}\sqrt 5 = \frac{a}{b}{/tex}
{tex}\Rightarrow {/tex}  b {tex}\sqrt 5 {/tex}=a
Squaring both sides, we get
{tex}\Rightarrow {/tex} 5b ² =a ² ... (1)
It means that 5 is factor of a²
Hence, 5 is also factor of a by Theorem. ... (2)
If, 5 is factor of a , it means that we can write a = 5c for some integer c .
Substituting value of a in (1) ,
5b² = 25c²
⇒ b² =5c²
It means that 5 is factor of b² .
Hence, 5 is also factor of b by Theorem. ... (3)
From (2) and (3) , we can say that 5 is factor of both a and b .
But, a and b are co-prime .
Therefore, our assumption was wrong. {tex}\sqrt 5 {/tex} cannot be rational. Hence, it is irrational.