Question

if p is a prime number which on dividing that √p is irrational
number​

Answer 1

ANSWER:

Let us assume, to the contrary, 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.