[Midwest Product]

Show n even if and only if n^2 is even.

[kowwok]

n==0?

[rms7936]

o is not even.
n=2?

[Brewer]

All even numbers remain even numbers after they are squared.
Therefore the answer is: all even numbers.

[Gloin]

He's asking for a proof, not further assumptions.

He wants you to prove that
even n -> even n²
and
even n² -> even n

I'll give it a try later.. Induction proof seems to be the way to go..

[Gloin]

Or actually..
1) Every even number n is divisible by 2 and can therefore be written as 2*k where k is a natural number.
2) this means that the square of n can be written as (2*k)*(2*k) = 2*(2*k*k) which is obviously even based on 1.

3) Further, any even square must be a multiple of 4 since the sqrt 2 is irrational and can never become a natural number by multiplication.
4) Therefore any even square can be written as 4*k*k = 2*2*k*k = (2*k)*(2*k). Taking the sqrt of (2*k)*(2*k) yields 2*k which again is even by 1.

Proof concluded. (maybe)

This post has been edited by Gloin: 05 March 2010 - 12:43 PM