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If P is a prime and x is a real number such that x^2 = P, then x is irrational.
Solution
Let, x be rational
So, x=r/s ,r,s integers, gcd(r,s)=1
SO,
(r/s)^2=P
r^2=Ps^2
Henec, P|r^2
P is prime so , P|r
Hence, r=kP for some integer k
k^2P^2=Ps^2
Pk^2=s^2
So, P|s
s=tP for some integer t
But, gcd(r,s)=1
HEnce a contradiction
HEnce, x is irrational

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  • 1. If P is a prime and x is a real number such that x^2 = P, then x is irrational. Solution Let, x be rational So, x=r/s ,r,s integers, gcd(r,s)=1 SO, (r/s)^2=P r^2=Ps^2 Henec, P|r^2 P is prime so , P|r Hence, r=kP for some integer k k^2P^2=Ps^2 Pk^2=s^2 So, P|s s=tP for some integer t But, gcd(r,s)=1 HEnce a contradiction HEnce, x is irrational