Question

For every prime number p prove that p is irrational

Answer 1

Correct Question:

If p is a prime number, then prove that √p is irrational.

Let us assume, that √p is rational.

So, we can find coprime integers a and b(b ≠ 0)

such that

√p = a/b => √p b = a => pb2 = a2 ….(i)

[Squaring both the sides] => a2 is

divisible by p => a is divisible by p So, we can write a = pc for some integer c.

Therefore, a2 = p2c2 ….[Squaring both the sides]

=> pb2 = p2c2 ….[From (i)]

=> b2 = pc2

=> b2 is divisible by p

=> b is divisible by p

=> p divides both a and b.

=> a and b have at least p as a common factor. But this contradicts the fact that a and b are coprime.

This contradiction arises because we have assumed that √p is rational.

Therefore, √p is irrational.

Hope it helps you ❣️☑️