Prove that √5 is irrational.

It’s a little tricky one🤔,

Let √5 is a rational number.
Therefore, we can find two integers a, b (b ≠ 0) such that √5 = a/b Let a and b have a common factor other than 1. Then we can divide them by the common factor, and assume that a and b are co-prime.
a = √5b
⇒ a² = 5b²
Therefore, a² is divisible by 5 and it can be said that a is divisible by 5.
Let a = 5k, where k is an integer
(5k)² = 5b²
⇒ 5k² = b²
This means that b² is divisible by 5 and hence, b is divisible by 5.
This implies that a and b have 5 as a common factor.
And this is a contradiction to the fact that a and b are co-prime.
Hence, √5 cannot be expressed as p/q or it can be said that √5 is irrational.

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