Question

show that √n is irrational if n is prime or not a perfect square ​

Answer 1

Let, a/b are co- prime

taking square both side ,we get

=> n = a^2/b^2

=> nb^2 = a^2 ......(1)

so, n divide a^2

it means n also divide a

for some integer c

a = nc

now squaring both side

a^2 = n^2c^2

=> nb^2 = n^2c^2 [ from (1) ]

=> b^2 = nc^2

so , n divide b^2

it means b also divide b

so, a and b have n as a prime factor

but this contradict the fact that a and b are co- prime .

therefore , our assumption is wrong .

hence, √n is irrational

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