prove there is no integer solution to x^2 -6 = 0 by contradiction

Question

Prove "There is no integer solution to the equation x^2 -6 =0." by contradiction.

I have no idea for this, can I juz move the -6 to right-hand side and square root both side to prove it? But it seems like not the correct way because the question hv 6 marks

Someone please help me

Answer 1

Assume an integer solution $x$ such that $x^2 = 6$ exists. An integer is either odd or even.

The square of odd $x$ is odd (whereas $6$ is even), so $x$ must be even. Let $x = 2k$, where $k$ is an integer.

Then $4k^2 = 6$. The LHS is divisible by $4$ but the RHS isn't.

Contradiction. Hence an integer solution cannot exist.