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

1.	If p is a prime number then prove that √p is irrational
Answer» Let us assume, to the contrary, that √p\xa0isrational.So, we can find coprime integers a and b(b ≠\xa00)such that √p = a/b=>\xa0√p b = a=> pb2\xa0= a2\xa0….(i) [Squaring both the sides]=> a2\xa0is divisible by p=> a is divisible by pSo, we can write a = pc for some integer c.Therefore, a2\xa0= p2c2\xa0….[Squaring both the sides]=>\xa0pb2\xa0= p2c2\xa0….[From (i)]=> b2\xa0= pc2=> b2\xa0is 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 haveassumed that √p is rational.Therefore,\xa0√p is irrational.

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