91Ó°ÊÓ

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(x, y \in \mathbb{R} .\) Then \(x^{3}+x^{2} y=y^{2}+x y\) if and only if \(y=x^{2}\) or \(y=-x\).

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). There exists a positive real number \(x\) for which \(x^{2}<\sqrt{x}\).

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(a, b \in \mathbb{Z}\). Prove that \(a+b\) is even if and only if \(a\) and \(b\) have the same parity.

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). There is a prime number between 90 and 100 .

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). Suppose \(a, b\) and \(c\) are integers. If \(a \mid b\) and \(a \mid\left(b^{2}-c\right),\) then \(a \mid c\).