# Prove that √5 is irrational

•Let x and y have a common factor other than 1, and so we can divide by that common factor and assume that x and y are coprime. So, y√5 = x.

•Squaring both side, we get, 5 y^2 = x^2.

•Thus, x^2 is divisible by 5, and by theorem we can say that x is also divisible by 5.

•Hence, x = 5z for some integer z.

•Substituting x, we get, 5x^2 = 25 z^2 i.e. y^2 = 25 z^2; which means y^2 is divisible by 5, and so y will also be divisible by 5.

•Now, from theorem, x and y will have 5 as a common factor. But, it is opposite to fact that x and y are co-prime.

•Hence, we can conclude √5 is irrational.