Question

If a square number is even then so is its square root.

Answer 1

If
`n^2` is even, then so is n.

Explanation:

We will prove the above theorem with the help of contrapositivty.

The contrapositive statement for given statement is, if n is odd then,
`n^2` is odd. Now, all we need to prove is for odd n we have odd
`n^2`.

Let n be odd, then it can be written in the form
`n = 2t + 1`, where t is an integer,

`n^2 = (2t + 1)^2 = 4t^2 + 1 + 4t = 2(2t^2 + 2t) + 1`

`n^2` is odd as it can be expressed in the form
`n^2 = 2s + 1, \text{ where s =}~ 2t^2 + 2t}`.

Hence, by contrapositivity, it can be said that if a number is even, then so is its square root.

But the given statement is true for
`m > 2`.