Arithmetic of parity
Theorem 4.3 (Arithmetic of parity). Let \(m, n \in \Z \).
- (a)
- If \(m\) and \(n\) have the same parity, then \(m + n\) is even. If they have different parities, then \(m + n\) is odd.
- (b)
- If \(m\) is even, then \(mn\) is even. If \(m\) and \(n\) are both odd, then \(mn\) is odd.
Proof. Write the even numbers as \(2a\) and the odd ones as \(2a + 1\).
- (a)
- \(2a + 2b = 2(a + b)\) and \((2a + 1) + (2b + 1) = 2(a + b + 1)\) are even, while \(2a + (2b + 1) = 2(a + b) + 1\) is odd.
- (b)
- \((2a)n = 2(an)\) is even, and \((2a + 1)(2b + 1) = 2(2ab + a + b) + 1\) is odd.