Question

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

Answer 1

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If possible,let √p be a rational number.

also a and b is rational.

then,√p = a/b

on squaring both sides,we get,

(√p)²= a²/b²

→p = a²/b²

→b² = a²/p [p divides a² so,p divides a]

Let a= pr for some integer r

→b² = (pr)²/p

→b² = p²r²/p

→b² = pr²

→r² = b²/p [p divides b² so, p divides b]

Thus p is a common factor of a and b.

But this is a contradiction, since a and b have no common factor.

This contradiction arises by assuming √p a rational number.

Hence,√p is irrational.

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