Question

Prove that 5`sqrt(2)` is irrational.

Answer 1

Let 5`sqrt(2)` is a rational number.
∴ 5`sqrt(2) = p/q` , where p and q are some integers and HCF(p, q) = 1     …(1)
⇒5√2q = p
⇒(5`sqrt (2)  q)^2 = p^2`
⇒` 2(25q^2) = p^2`
⇒ `p^2 `is divisible by 2
⇒ p is divisible by 2 ….(2)
Let p = 2m, where m is some integer.
∴5`sqrt(2)` 𝑞 = 2m
⇒(5`sqrt(2) q )^2 = (2m)^2`
⇒` 2(25 q^2) = 4m^2`
⇒`25q ^2 = 2m^2`
⇒` q^ 2 `is divisible by 2
⇒ q is divisible by 2 ….(3)
From (2) and (3) is a common factor of both p and q, which contradicts (1).
Hence, our assumption is wrong.
Thus, 5`sqrt(2)` is irrational.