91Ó°ÊÓ

Use the method of direct proof to prove the following statements.. If \(x\) is an even integer, then \(x^{2}\) is even.

Use the method of direct proof to prove the following statements. Suppose \(x, y \in \mathbb{Z} .\) If \(x\) is even, then \(x y\) is even.

Use the method of direct proof to prove the following statements. If \(x \in \mathbb{R}\) and \(0

Use the method of direct proof to prove the following statements. If two integers have the same parity, then their sum is even. (Try cases.)

Use the method of direct proof to prove the following statements. Suppose \(x\) and \(y\) are positive real numbers. If \(x

Use the method of direct proof to prove the following statements. If \(n \in \mathbb{N}\) and \(n \geq 2,\) then the numbers \(n !+2, n !+3, n !+4, n !+5, \ldots, n !+n\) are all composite. (Thus for any \(n \geq 2,\) one can find \(n-1\) consecutive composite numbers. This means there are arbitrarily large "gaps" between prime numbers.)

Use the method of direct proof to prove the following statements. Suppose \(a, b \in \mathbb{N}\). If \(\operatorname{gcd}(a, b)>1\), then \(b \mid a\) or \(b\) is not prime.