Contrapositive Proof of Even Squares

Prove that if $n^2$ is even, then $n$ is even using a proof by contrapositive.

In this problem, you're tasked with proving a fundamental result about integers using a proof by contrapositive. Proof by contrapositive is a common proof technique in mathematics. Instead of proving a direct implication 'if P then Q', you prove the contrapositive 'if not Q then not P'. These are logically equivalent: proving one proves the other. Here, the direct statement is that if the square of an integer is even, then the integer itself must be even. Its contrapositive would be: if an integer is odd, then its square is odd. The concept of even and odd integers is central to this proof. An integer is even if it can be expressed as 2 times some integer, and odd if it can be expressed as 2 times an integer plus one. Showing how squaring an odd number results in another odd number can elegantly prove the statement by contrapositive. This proof strategy is widely applicable in discrete mathematics and helps to reinforce an understanding of logical equivalences and implications. By practicing this kind of proof, you're also deepening your understanding of number properties, particularly in the realms of parity (oddness and evenness) and their implications in mathematical reasoning.