Question

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

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Answer 1

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Answer 2

❥︎❥︎Answer꧂

Let p be a rational number and p= a/b

➪p=a^2/b^2

➪a^2=pb^2

➪p divides a^2

But when a prime number divides the product of two numbers, it must divide atleast one of them.

here a^2 =a×a

p divides a

Let a=pk

(pk)^2 =pb^2

➪p^2k^2

➪b^2=pk^2

∴p divides b^2

But b^2=b×b

∴p divides b

Thus, a and b have atleast one common multiple p.

But it arises the contradiction to our assumption that a and b are co-prime.

Thus, our assumption is wrong and p is irrational number.

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