Prove that for any prime positive integer `p ,sqrt(p)` is an irrational number.

Explanation

Let p be a prime number and if possible let `sqrt(p)` be irrational.
Let the simplest from of `sqrt(p) be (a)/(b).`
Then a and b are integers and having no common factors other than 1 and `b ne 0`.
Now, `sqrt(p) = (a)/(b)`
`rArr` `p = (a^(2))/(b^(2))`
`rArr` `a^(2) = pb^(2)" "`...(1)
As `pb^(2)` is divisible by p.
`therefore a^(2)` is divisible by p.
`rArr` a is divisible by p.
Let a = pc for some integer c.
From equation (1)
`(pc)^(2) = pb^(2)`
`rArr` `b^(2) = pc^(2)`
But `pc^(2)` is divisible by p.
`therefore` `b^(2)` is divisible by p.
`rArr` b is divisible by p.
Let b = pd for some integer d.
Thus, p is a common factor of both a and b.
But it contradicts the fact that a and b have no common factor other than 1.
So, our supposition is wrong.
Hence, `sqrt(p)` is irrational. `" "` Hence Proved.