Q. 114.1( 16 Votes )

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Answer :

Given: Any prime positive integer p.


To find: is an irrational number.


Concept used: Assume p to be rational number and prove it is irrational by contradiction.


Explanation:


Let assume that √p is rational


Therefore, it can be expressed in the form of , where a and b are integers and b≠0


Therefore, we can write




a2 = pb2


Since a2is divided by b2, therefore a is divisible by b.


Let a = kc


(kc)2= pb2


K2c2= pb2


Here also b is divided by c, therefore b2 is divisible by c2. This contradicts that a and b are co-primes. Hence is an irrational number.


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