Question

prove that √5 is irrational.​

Answer 1

let us assume the contrary that √5 is rational.

That is we can find integers a and B (≠0) such that√5=a/b

suppose a and B have a common factor other than 1, then we can divide by the common factor and assume that a and b are coprime.

so b√5=a

squaring on both sides we get 5b²=a²

therefore a²is divisible by 5.

so we can write a=5c for some integer c.

we get 5b²=25c²

that is b²=5c²

this means b² is divisible by 5 and so b is also divisible by 5.

therefore a and b have atleast 5 as a common factor.

this contradicts the fact that a and b are coprime.

This conclude √5 as irrational.