Let \(x\) and \(y\) be integers. Prove that if \(xy\) is odd, then \(x\) and \(y\) are both odd.

If \(m^2\) is odd, then \(m\) is odd.

If \(n^3 + 5\) is odd, then \(n\) is odd.

If \(n^2\) is even, then \(n\) is even.

If \(7x + 9\) is even, then \(x\) is odd.

Suppose \(a\), \(b\), and \(n\) are positive integers. If \(n = ab\), then either \(a \le \sqrt{n}\) or \(b \le \sqrt{n}\).

If \(5x - 7\) is even, then \(x\) is odd.

If \(3 \mid a^2,\) then \(3 \mid a.\)