Theorem 4.4 (Parity of a square). For \(n \in \Z \), \(n^2\) is even if and only if \(n\) is even.
Proof.If \(n\) is even, then \(n^2 = n \cdot n\) is even by Theorem 4.3. For the converse we prove the contrapositive
(Theorem 1.3): if \(n\) is not even, then by Theorem 4.2 it is odd, so \(n^2\) is odd by Theorem 4.3, and
hence not even, again by Theorem 4.2.